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The article considers a linear consecutive-k-out-of-n: F system and a linear connected-(r, s)-out-of-(m, n): F system. It is assumed that the components are identical and the failure times of components follow an exponential distribution. The system is continuously monitored and the component failure can be known at any time. A preventive maintenance policy is proposed, and it is based on the system critical condition which is related to the number of working components in the minimal cut sets of the system. If there is at least one minimal cut set which consists of only one working component, the system is maintained preventively after a certain time interval and the failed components are replaced with the new ones to prevent the system failures. The expected cost rate is used as an optimization criterion. The expected cost rate is estimated by simulation and numerical examples are studied. The numerical analyses are exploratory and not generalizable.
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Special Issue Article
Proc IMechE Part O:
J Risk and Reliability
1–8
ÓIMechE 2015
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DOI: 10.1177/1748006X15588249
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A preventive replacement policy based
on system critical condition
Won Young Yun and Alfonsus Julanto Endharta
Abstract
The article considers a linear consecutive-k-out-of-n: F system and a linear connected-(r,s)-out-of-(m,n): F system. It is
assumed that the components are identical and the failure times of components follow an exponential distribution. The
system is continuously monitored and the component failure can be known at any time. A preventive maintenance policy
is proposed, and it is based on the system critical condition which is related to the number of working components in
the minimal cut sets of the system. If there is at least one minimal cut set which consists of only one working compo-
nent, the system is maintained preventively after a certain time interval and the failed components are replaced with the
new ones to prevent the system failures. The expected cost rate is used as an optimization criterion. The expected cost
rate is estimated by simulation and numerical examples are studied. The numerical analyses are exploratory and not
generalizable.
Keywords
Condition-based maintenance, multicomponent, consecutive-k-out-of-nsystem, connected-(r,s)-out-of-(m,n) system,
reliability, simulation
Date received: 30 September 2014; accepted: 22 April 2015
Introduction
Preventive maintenance (PM) problems of two systems
with multicomponents, consecutive-k-out-of-n: F sys-
tem and a connected-(r,s)-out-of-(m,n): F system, are
considered. Consecutive-k-out-of-n: F systems, which
consist of ncomponents arranged linearly or circularly,
fail if there are kconsecutive failed components.
1
Consecutive-k-out-of-n: F systems are commonly used
in integrated circuits, microwave relay stations in tele-
communications, oil pipeline system, vacuum system in
accelerators, computer ring networks (kloop), and
spacecraft relay stations.
2
Most research related to consecutive-k-out-of-n:F
systems investigated how to calculate the system relia-
bility and some articles deal with the system design
problems to obtain the optimal arrangement of compo-
nents and to determine the optimal values of kand/or
n. There is as yet little research into maintenance poli-
cies for the consecutive-k-out-of-n: F system. The relia-
bility of consecutive-k-out-of-n: F systems can be
calculated from the recursive equation in Bollinger and
Salvia.
3
The system reliabilities of linear and circular
consecutive-k-out-of-n: F systems were also proposed
by Lambiris and Papastavridis.
4
Pekoz and Ross
5
sim-
plified the Lambiris and Papastavridis’ reliability
formulas. Canfield and McCormick
6
proposed an
asymptotic reliability formulation. Yun et al.
7
proposed
a modified formula to calculate the system reliability of
a circular consecutive-k-out-of-n: F system and a
method to find the optimal system design for the circu-
lar consecutive-k-out- of-n: F system with (k21)-step
Markov dependence.
Flynn and Chung
8
proposed a maintenance policy
related to the critical component policies (CCPs) for
consecutive-k-out-of-n: F systems. The failed compo-
nents are replaced only if the components are in the
critical component set. They used a branch and bound
algorithm to find the optimal CCP. Zuo and Wu
9
stud-
ied an age replacement policy (age PM) for k-out-of-n:
F system and consecutive-k-out-of-n: F system. Yun
et al.
10
considered system design and maintenance prob-
lems for linear and circular consecutive-k-out-of-n:F
systems with load sharing dependence. An age PM is
used to maintain preventively in Yun et al.
10
Department of Industrial Engineering, Pusan National University, Busan,
Korea
Corresponding author:
Won Young Yun, Department of Industrial Engineering, Pusan National
University, Busan 609-735, Korea.
Email: wonyun@pusan.ac.kr
at PUSAN NATL UNIV on July 19, 2015pio.sagepub.comDownloaded from
Salvia and Lasher
11
considered systems with multidi-
mensional arrangements of components, such as two-
dimensional (2D) or three-dimensional (3D) systems.
According to Salvia and Lasher, 2D systems consist of
a square grid of size nwhich contains n
2
components
and 3D system consists of a cubic grid of size nwhich
contains n
3
components. The system fails if there is at
least a square or a cube of size kwith failed compo-
nents. Boehme et al.
12
studied a connected-(r,s)-out-of-
(m,n): F system. This system consists of components
arranged into a rectangular pattern with mrows and n
columns. The system fails, if there is at least one grid of
rrows and scolumns of which all components are failed
(1 4r4m,14s4n). Yun et al.
13,14
studied a system
design optimization problem in connected-(r,s)-out-of-
(m,n): F systems with identical and independent com-
ponents, where the model parameters (r,s) are decision
variables. Yun et al.
15
estimated the system reliability of
the linear consecutive-(r,s)-out-of-(m,n): F lattice sys-
tem with failure dependence by simulation and pro-
posed a PM policy based on age.
In this article, we consider a PM problem of multi-
component systems. A condition-based maintenance
(CBM) policy is proposed. A critical condition is
defined to maintain the system preventively. If the criti-
cal condition occurs, which is when there is one mini-
mal cut set consisting of one working component, the
system will be maintained preventively after a certain
amount of time. If the system fails before the predeter-
mined time interval, the system is maintained correc-
tively. All failed components are replaced at
maintenance times and replacement cost is considered.
The expected cost rate is used as the optimization cri-
terion and is estimated by simulation. The predeter-
mined PM time interval (T
PM
) is obtained to minimize
the expected cost rate.
The article is organized as follows. Section
‘‘Maintenance policy’’ shows the maintenance policy
studied in the article. In section ‘‘Methodology,’’ the
assumptions are described and the simulation proce-
dure for estimating the expected cost rate is explained.
We show the numerical examples in section ‘‘Numerical
examples.’’ Section ‘‘Conclusion’’ concludes the article.
Maintenance policy
Corrective maintenance policy
Corrective maintenance policy is the simplest policy. In
corrective maintenance policy, the system is maintained
only at system failure. Therefore, this policy brings
high unavailability when the maintenance processing
time or the replacement time is not negligible. The
expected cost rate EC
CM
yielded with this policy can be
calculated by equation (1). In this policy, the cost para-
meters included in the expected cost rate are the cost of
CM c
CM
and the cost of component replacement c
R
.
The expected lifetime of the system, L
CM
, is the mean
time to the system failure (MTTF) and Nis the
expected number of failed components at the system
failure time
ECCM =cCM +cR3N
LCM
ð1Þ
Age-based maintenance policy
Age-based maintenance policy is a well-known policy.
The system is maintained preventively at time T
A
or cor-
rectively at system failure, whichever occurs first.
16
In
order to find the minimum expected cost rate EC
A
(T
A
),
we need to optimize the maintenance policy parameter:
age T
A
. The cost parameters in the age-based mainte-
nance policy are the cost of CM c
CM
, the cost of PM
c
PM
, and the cost of component replacement c
R
.The
expected lifetime of the system, L
A
(T
A
), is the MTTF or
to the PM time T
A
,R
S
(T
A
) is the system reliability at
time T
A
,andN
A
(T
A
) is the expected number of failed
components at the system failure or at time T
A
.The
expected cost rate can be calculated as follows
ECA(TA)=
cCM 31RS(TA)ðÞ+cPM 3RS(TA)+cR3NA(TA)½
LA(TA)
ð2Þ
CBM policy
CBM policy is the new proposed policy. The system
condition is continuously monitored and the component
failure can be known at any time. The maintenance pol-
icy is proposed based on the system criticality condition.
We derive the system criticality condition related to the
number of working component in the system minimal
cut sets. If there is at least one minimal cut set which is
consisting of one working component, the system is
maintained preventively after a certain time interval
T
PM
. If the system failure occurs before reaching time
point T
PM
, the system is maintained correctively.
Consider a consecutive 3-out-of-5: F system as an
example. In this example, there are three minimal cut
sets and each minimal cut set consists of three compo-
nents. Figure 1 shows some examples of component
failure paths, where ‘1’ represents the working compo-
nent and ‘0’ represents the failed component. In
Figure 1(a), after the failure of the second component,
the first minimal cut set contains only one working
component and PM will be done after T
PM
time inter-
val. While Figure 1(b) and (c) shows that after a certain
time interval T
PM
from the third component failure
time, the system will be maintained preventively. In
Figure 1(b), the third failed component (which is the
second component) makes the first minimal cut set have
only one working component. While in Figure 1(c), the
third failed component (which is the third component)
makes two minimal cut sets have one working compo-
nent at the same time. Figure 1(d) shows that there is
2Proc IMechE Part O: J Risk and Reliability
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another component failure which leads to the system
failure before reaching PM time, so the system is main-
tained correctively.
The cost parameters included in this maintenance
policy are the cost of CM c
CM
, the cost of PM c
PM
,
and the cost of component replacement c
R
.R
S
(T
PM
)is
the probability that the system will be maintained pre-
ventively regarding the system critical condition and
time interval T
PM
.L
Cr
(T
PM
) is the expected time to the
renewal time, which is the system failure time or the
PM time. N
Cr
(T
PM
) is the expected number of failed
components at the renewal time
ECCr(TPM )=
cCM 31RS(TPM)ðÞ+cPM 3RS(TPM)+cR3NCr(TPM )½
LCr(TPM )
ð3Þ
Methodology
Assumptions
1. Components and system have two states: working
and failed.
2. States of the components are monitored continu-
ously, with no cost.
3. Replacement times are negligible.
4. Component failure times are independent and iden-
tically exponentially distributed.
Simulation procedure
The objective of the problem is to find the optimal
maintenance policy which minimizes the expected
cost rate. However, it is difficult to obtain the closed
form of the expected cost rate in section ‘‘Maintenance
policy.’’ Therefore, we estimate the expected cost
rate by simulation. The simulation procedure is as
follows:
Step 1. Initialize the simulation and cost variables, such
as simulation number (SRN); total cycle time; total
number of component failures; the number of working
system at the end of simulation; and cost of CM, PM,
and component replacement.
Step 2. Initialize the system variables, such as the state
of component in the system, system state, number of
failed component, and lifecycle time.
Step 3. Generate the interfailure times of working com-
ponents based on the component failure time distribu-
tion, which is exponential distribution with
parameter l.
Step 4. Choose the next failed component and check
system condition or number of working components in
each minimal cut set (for the CBM policy), then decide
whether the system is maintained correctively or pre-
ventively. If system failure occurs, go to Step 5.
Otherwise, repeat Step 4.
Step 5. Terminate simulation if the termination condi-
tion is satisfied. Otherwise, go to Step 2. In this article,
the termination condition is the number of simulation
replication (SRN = 100).
Step 6. Summarize output and estimate the expected
cost rate.
The detailed procedures for three maintenance poli-
cies for consecutive-k-out-of-n: F systems are noted in
Appendices 1–3. For connected-(r,s)-out-of-(m,n): F
system, some changes are necessary. All possible T
A
and T
PM
are enumerated for finding the optimal T
A
and T
PM
.
Numerical examples
Numerical examples are studied for a linear consecu-
tive-k-out-of-n: F system and a linear connected-(r,s)-
out- of-(m,n): F system. The numerical examples will
show the effects of cost and system parameters on the
optimal time interval PM. It is assumed that component
failures are independent and follow an exponential dis-
tribution with parameter l.
Linear consecutive-k-out-of-n: F system
Numerical studies are done in cases with different val-
ues of system parameters, such as k= {3, 5}, and
n= {5, 10, 15, 20}. The replication number in simula-
tion (SRN) is 100 and we set C
R
= 1 and c
PM
= 2 with
the component failure rate l= 0.02 as the basis. The
simulation results in cases with various values of para-
meters are shown in Table 1. An age PM and corrective
replacement policy (CM) are also estimated by simula-
tion and the expected cost rates are compared to evalu-
ate the goodness of the CBM policy. The computation
Figure 1. Illustration of the proposed maintenance policy
based on the system critical condition. (a) PM occurs after the
second component failure and there is one minimal cut set with
one working component, (b) PM occurs after the third
component failure and there is one minimal cut set with one
working component, (c) PM occurs after the third component
and there are two minimal cut sets with one working
component, and (d) CM occurs.
Yun and Endharta 3
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time of getting the expected cost rate with CBM policy
by using the proposed simulation procedure is less than
5 s for consecutive-5-out-of-20: F system. T
A
is the time
to PM for the age PM policy; EC
CBM
,EC
AgePM
, and
EC
CM
are the expected cost rates for the CBM policy,
the age PM policy, and the CM policy, respectively.
s
CBM
,s
AgePM
, and s
CM
are the standard deviations of
the expected cost rates for the CBM policy, the age PM
policy, and the CM policy, respectively. The optimal
decision variables (T
PM
and T
A
) are found by
enumeration.
Table 1 shows that if the number of minimal cut sets
(n2k+ 1) increases, T
PM
increases because the prob-
ability of the system failure decreases. If the ratio of
CM cost to PM cost increases, T
PM
becomes shorter.
To evaluate the performance of the CBM policy, 95%
confidence interval plots are used. Figure 2 shows the
evaluation results for a linear consecutive-3-out-of-5: F
system. When the ratio of the CM cost to the PM cost
is less than 5, the CBM policy does not give a different
result from the age PM and CM policies significantly
because the expected cost rate of the CBM policy is
within the 95% confidence intervals of the other poli-
cies. But when the ratio of the CM cost to the PM cost
is 5, the CBM policy outperforms age PM and CM pol-
icies because the expected cost rate of the CBM policy
is less than the lower bound of the confidence interval
of the other policies. It is shown that for the small ratio
of CM cost to PM cost (c
CM
/c
PM
= 2), the expected
cost rates are same for all policies and for any system
size based on the 95% confidence interval plot in
Figure 3. The CBM policy outperforms the other poli-
cies only for the linear consecutive-3-out-of-5: F system
for the case with c
CM
/c
PM
= 5, while for the other sys-
tems, the expected cost rate from the CBM policy is
not different from the expected cost rates from the
other policies significantly based on the 95% confi-
dence interval plot in Figure 4.
Linear connected-(r, s)-out-of-(m, n): F system
Numerical examples for a 2D system are presented for
linear connected-(3,3)-out-of-(5,5): F system. The
Table 1. Effect of system parameters in linear consecutive-k-out-of-n: F systems and the comparison of CBM policy with the age PM
and CM.
Parameter CBM Age PM CM
c
CM
/c
PM
kn T
PM
EC
CBM
s
CBM
T
A
EC
AgePM
s
AgePM
EC
CM
s
CM
2 3 5 439 0.0544 0.0032 504 0.0546 0.0029 0.0557 0.0036
10 601 0.0795 0.0042 778 0.0791 0.0056 0.0802 0.0050
15 923 0.0929 0.0064 1380 0.0926 0.0044 0.0955 0.0056
5 10 416 0.0448 0.0014 646 0.0450 0.0017 0.0449 0.0016
15 517 0.0506 0.0014 803 0.0508 0.0017 0.0517 0.0022
20 634 0.0550 0.0014 1111 0.0552 0.0021 0.0560 0.0022
3 3 5 178 0.0692 0.0045 312 0.0690 0.0036 0.0694 0.0053
10 217 0.0969 0.0048 874 0.0953 0.0048 0.0978 0.0042
15 343 0.1099 0.0057 1119 0.1101 0.0056 0.1122 0.0082
5 10 97 0.0506 0.0026 374 0.0523 0.0024 0.0534 0.0025
15 297 0.0574 0.0025 683 0.0577 0.0021 0.0586 0.0022
20 430 0.0617 0.0024 949 0.0615 0.0015 0.0630 0.0029
4 3 5 38 0.0753 0.0050 279 0.0824 0.0047 0.0832 0.0049
10 80 0.1118 0.0108 641 0.1124 0.0048 0.1147 0.0061
15 179 0.1288 0.0056 842 0.1274 0.0041 0.1303 0.0101
5 10 37 0.0525 0.0021 227 0.0577 0.0026 0.0605 0.0024
15 119 0.0629 0.0032 411 0.0641 0.0027 0.0659 0.0026
20 228 0.0670 0.0030 524 0.0678 0.0021 0.0695 0.0026
5 3 5 21 0.0778 0.0068 109 0.0954 0.0070 0.0971 0.0055
10 72 0.1238 0.0087 137 0.1257 0.0094 0.1312 0.0059
15 105 0.1439 0.0078 173 0.1411 0.0070 0.1460 0.0067
5 10 42 0.0531 0.0029 198 0.0644 0.0038 0.0690 0.0027
15 80 0.0661 0.0037 247 0.0709 0.0034 0.0729 0.0021
20 138 0.0716 0.0026 306 0.0742 0.0035 0.0758 0.0031
CBM: condition-based maintenance; age PM: age replacement policy; CM: corrective replacement policy.
Figure 2. The 95% confidence interval plot for expected cost
rate for linear consecutive-3-out-of-5: F system.
4Proc IMechE Part O: J Risk and Reliability
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system has (5 23 + 1)(5 23 + 1) = 9 minimal cut
sets. As a basis, we set C
R
= 1 and c
PM
= 2 with the
component failure rate l= 0.02. The simulation result
is shown in Table 2. The optimal T
PM
and T
A
are
found by using enumeration. The optimal T
PM
becomes larger if the ratio of the CM cost to the PM
cost increases. Figure 5 shows the 95% confidence
interval for the expected cost rates for the linear
connected-(3,3)-out-of-(5,5): F system. Based on Table
2 and Figure 5, if the ratio of CM cost to PM cost is
larger than 5, the CBM policy outperforms the other
policy because the expected cost rate from the CBM
policy is less than the lower bounds of the other poli-
cies. However, if the ratio of the CM cost to the PM
cost is less than 6, the performance of the CBM policy
is similar to the other policies.
Conclusion
In this article, we considered a PM problem in a linear
consecutive-k-out-of-n: F system and a linear con-
nected-(r,s)-out-of-(m,n): F system. A CBM policy is
proposed. The system critical condition is monitored
continuously. If the observed system reaches the critical
condition, which is when there is one minimal cut set
consisting of only one working component, we main-
tain the system preventively by replacing the failed
components by new ones after a certain predetermined
time interval. If the system fails before reaching the pre-
ventive replacement time, all failed components are
replaced correctively. The decision variable in the CBM
policy is the preventive replacement time interval, T
PM
.
The expected cost rate is used to determine the optimal
T
PM. We use simulation to estimate the expected cost
rate. Some numerical examples were studied. The
Figure 3. The 95% confidence interval plot for expected cost
rate for the system with c
CM
/c
PM
=2.
Figure 4. The 95% confidence interval plot for expected cost
rate for the system with c
CM
/c
PM
=5.
Table 2. Effect of system parameters in linear connected-(3,3)-out-of-(5,5): F system and the comparison of CBM policy with the
age PM and CM.
c
CM
/c
PM
CBM Age PM CM
T
PM
EC
CBM
s
CBM
T
A
EC
AgePM
s
AgePM
EC
CM
s
CM
2 756 0.0339 0.0012 1663 0.0335 0.0029 0.0337 0.0025
3 562 0.0368 0.0014 1466 0.0363 0.0023 0.0375 0.0017
4 192 0.0381 0.0019 1259 0.0387 0.0021 0.0397 0.0012
5 55 0.0398 0.0034 895 0.0409 0.0033 0.0431 0.0019
6 53 0.0398 0.0022 746 0.0438 0.0028 0.0448 0.0016
7 39 0.0398 0.0017 589 0.0454 0.0022 0.0481 0.0029
CBM: condition-based maintenance; age PM: age replacement policy; CM: corrective replacement policy.
Figure 5. The 95% confidence interval plot for expected cost
rate for linear connected-(3,3)-out-of-(5,5): F system.
Yun and Endharta 5
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numerical analyses are exploratory and not generaliz-
able. The optimal T
PM
is found by using enumeration
method. The simulation result shows that if the ratio of
the CM cost to PM cost increases, the optimal T
PM
becomes shorter. If the ratio of the CM cost to the PM
cost is small, the expected cost rates from the proposed
replacement, age replacement, and the corrective main-
tenance policies are relatively same. However, if ratio
of the CM cost to the PM cost is high, the CBM policy
could outperform the other policies for cases with
smaller system size. For further studies, we will obtain
the expected cost rate analytically and find the optimal
solutions of decision variable of the CBM policy.
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
Funding
This research was supported by Basic Science Research
Program through the National Research Foundation
of Korea (NRF) funded by the Ministry of Education,
Science and Technology (NRF-2013R1A1A2060066).
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Appendix 1
Detailed procedure for CM policy
1. Initialize simulation variables:
SN =0,TCL =0, TFN =0, NSW =0
SN: simulation number, TCL: total cycle time,
TFN: total number of failed components, and
NSW: number of working system at the end of
simulation.
2. Initialize system variables:
s
i
= 1 for all i= 1,..., n,
t
i
=Nfor all i= 1,..., n,
FN =0,CL =0, s
sys
=1,
s
i
: the state of ith component, t
i
: the time to failure
of the corresponding component,
FN: number of failed components, CL: cycle time,
and
s
sys
: the system state.
3. Generate the component failure times:
For all i= 1,..., n
t
i
=2log(Uniform(0,1))/l
4. Get the system failure time:
w: smallest number of working components in
minimal cut set
While w.0
NF ={ijt
i
= min(t
j
), j= 1,..., n}
NF shows that the ith component will fail,
s
NF
=0,CL =CL +t
NF
,t
NF
=N.
For each minimal cut set
w
new
= total working component in the
minimal cut set
If w
new
\w, then
w=w
new
s
sys
=0,
FN =n2number of working components in the
system.
6Proc IMechE Part O: J Risk and Reliability
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5. Terminate simulation:
TCL =TCL + CL, TFN =TFN +FN
NSW =NSW +s
sys
,SN =SN +1,
If SN \SRN, then
Go to Step 2
Else, go to Step 6
6. Summarize output:
L
CM
=TCL/SRN,N=TFN/SRN, and
ECCM =cCM +cR3N
LCM
Appendix 2
Detailed procedure for age-based maintenance policy
1. Initialize simulation variables:
SN =0,TCL =0, TFN =0, NSW =0
SN: simulation number, TCL: total cycle time,
TFN: total number of failed components, and
NSW: number of working system at the end of
simulation.
2. Initialize system variables:
s
i
= 1 for all i= 1,...,n,
t
i
=Nfor all i= 1,..., n,
FN =0,CL =0, s
sys
=1,
s
i
: the state of ith component, t
i
: the time to failure
of the corresponding component,
FN: number of failed components, CL: cycle time,
and
s
sys
: the system state.
3. Generate the next failure:
For all i= 1,..., n
t
i
=2log(Uniform(0,1))/l
4. Get the system renewal time:
w: smallest number of working components in
minimal cut set
While w.0
If T
A
\CL + min(t
i
), then
s
sys
=1,CL =T
A
,t
NF
=N,
Go to Step 5
Else
NF ={ijt
i
= min(t
j
), j= 1,..., n}
NF shows that the ith component will fail
s
NF
=0,CL =CL +t
NF
,t
NF
=N.
For each minimal cut set
w
new
= total working component in the
minimal cut set
If w
new
\w, then
w=w
new
If w= 0, then s
sys
=0
FN =n2number of working components in the
system.
5. Terminate simulation:
TCL =TCL + CL, TFN =TFN +FN,
NSW =NSW +s
sys
,SN =SN +1,
If SN \SRN, then
Go to Step 2
Else, go to Step 6
6. Summarize output:
L
A
(T
A
)=TCL/SRN,N
A
(T
A
)=TFN/SRN,
R
S
(T
A
)=NSW/SRN, and
ECA(TA)=
cCM 31RS(TA)ðÞ+cPM 3RS(TA)+cR3NA(TA)½
LA(TA)
Appendix 3
Detailed procedure for conditional-based
maintenance policy
1. Initialize simulation variables:
SN =0,TCL =0, TFN =0, NSW =0
SN: simulation number, TCL: total cycle time,
TFN: total number of failed components, and
NSW: number of working system at the end of
simulation.
2. Initialize system variables:
s
i
= 1 for all i= 1,..., n,
t
i
=Nfor all i= 1,..., n,
FN =0,CL =0, s
sys
=1,
s
i
: the state of ith component, t
i
: the time to failure
of the corresponding component,
FN: number of failed components, CL: cycle time,
and
s
sys
: the system state.
3. Generate the next failure:
For all i= 1,..., n
t
i
=2log(Uniform(0,1))/l
4. Get the system renewal time:
w: smallest number of working components in
minimal cut set
While w.1
NF ={ijt
i
= min(t
j
), j= 1,..., n}
NF shows that the ith component will fail,
s
NF
=0,CL =CL +t
NF
,t
NF
=N.
For each minimal cut set
w
new
= total working component in the
minimal cut set
If w
new
\w, then
w=w
new
PMFrom = CL
PMFrom shows the starting time of T
PM
While w=1
If PMFrom + T
PM
\CL + min(t
i
), then
s
sys
=1,CL = PMFrom + T
PM
,t
NF
=N,
Go to Step 5
Else
NF ={ijt
i
= min(t
j
), j= 1,..., n}
NF shows that the ith component will fail,
s
NF
=0,CL =CL +t
NF
,t
NF
=N.
For each minimal cut set
w
new
= total working component in the
Yun and Endharta 7
at PUSAN NATL UNIV on July 19, 2015pio.sagepub.comDownloaded from
minimal cut set
If w
new
\w, then
w=w
new
If w= 0, then s
sys
=0
FN =n2number of working components in the
system
5. Terminate simulation:
TCL =TCL +CL,TFN =TFN +FN,
NSW =NSW +s
sys
,SN =SN +1,
If SN \SRN,then
Go to Step 2
Else, go to Step 6
6. Summarize output:
L
Cr
(T
PM
)=TCL/SRN,N
Cr
(T
PM
)=TFN/SRN,
R
S
(T
PM
)=NSW/SRN
ECCr(TPM )=
cCM 31RS(TPM)ðÞ+cPM 3RS(TPM)+cR3NCr (TPM)½
LCr(TPM )
8Proc IMechE Part O: J Risk and Reliability
at PUSAN NATL UNIV on July 19, 2015pio.sagepub.comDownloaded from
... Shen and Cui (2015) considered a circular consecutive-k-out-of-n: F system with sparsely connected IID components and developed some equations for system reliability and Birnbaum importance by means of the finite Markov chain imbedding approach. Yun and Endharta (2016) developed a condition-based PM policy for linear consecutive-k-out-of-n: F systems and linear connected-(r, s)-out-of-(m, n): F systems and used simulation to get the optimal decision variable. Endharta and Yun (2015) used simulation for periodic inspection policy. ...
... In this paper, we study a maintenance problem for circular consecutive-k-out-of-n: F systems. A condition-based PM policy which has been developed in Yun and Endharta (2016) is used, in which the system is maintained after a certain time interval if there are certain number of working components in the minimal cut sets. The unique case of the policy is considered so that the closed form of the model can be derived easily; that is if there is at least one minimal cut set with one working component, a preventive maintenance will be done after a certain time interval. ...
Purpose The purpose of this paper is to develop a preventive maintenance policy with continuous monitoring for a circular consecutive- k -out-of- n : F systems. A preventive maintenance policy is developed based on the system critical condition which is related to the number of working components in the minimal cut sets of the system. If there is at least one minimal cut set which consists of only one working component, the system is maintained preventively (PM) after a certain time interval and the failed components are replaced with the new ones to prevent the system failures. If the system fails prior to the preventive maintenance, the system is correctively maintained (CM) immediately by replacing the failed components. Design/methodology/approach The mathematical function of the expected cost rate for the proposed maintenance policy is derived. The costs of PM, CM, and replacement per component are considered. The optimal maintenance parameter, which is the PM interval, is obtained by enumeration, and the numerical studies are shown with various system and cost parameters. The performance of the proposed policy is evaluated by comparing its expected cost rate to those of the no-PM and age-PM policies. The percentage of cost increase from the no-PM and age-PM policies to the proposed PM policy is calculated and this value can represents how important the continuous monitoring in this policy. Findings The proposed policy outperforms other policies. When the cost of CM is high and the cost of PM is low, the proposed PM policy is more suitable. Research limitations/implications The system consists of identical components and the component failure times follow an exponential distribution. Continuous monitoring is considered, which means that the component states can be known at any time. Three cost parameters, cost of PM, CM, and replacement per component, are considered. Originality/value This paper shows a maintenance problem for circular consecutive- k -out-of- n : F systems. Many studies on this system type focused on the reliability estimation or system design problem. Previous study with this policy (Endharta and Yun, 2015) has been developed for linear systems, although the study used a simulation approach to estimate the expected cost rate. Also, Endharta et al. (2016) considered a similar method for the different types of system, which is linear consecutive- k -out-of- n : F system.
... The transitions from the beginning state into the end state (system failure state) is known as a system failure path. Yun et al. [3] and Yun and Endharta [4], [5], [6] used system failure paths to study the maintenance scheduling problem for other types of k-out-of-n systems. ...
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... An optimal age-based group maintenance policy is proposed for a multi-unit series system whose components are subject to different gradual degradation phenomena [11]. Replacement policies based on the critical conditions of a linear consecutive-k-out-of-n: F system and a linear connectedr s ( , )-out-ofm n ( , ): F system were proposed [12]. Imperfect preventive maintenance for the expired warranty was integrated with age replacement [13] and a warranty cost model for repairable products when age replacement policy is adopted in conjunction with the renewal of a minimal repair-replacement warranty was studied [14]. ...
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