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Optimal Preventive Maintenance for Repairable Weighted
k-out-of-nSystems
K. Hamdan,∗M. Tavangar†& M. Asadi‡
Abstract
Weighted k-out-of-nsystems form an important class of redundant systems with a
wide range of applications in reliability engineering. In this paper, we introduce novel
optimal preventive maintenance models for the class of weighted k-out-of-nsystems
based on average cost and availability criteria. Depending on the average number of
failed components, we set up a cost (availability) function corresponding to the time
that the total weights of the working components become less than a predetermined
threshold m, or the age of the system reaches TP M . The form of the cost (and avail-
ability) function relies on the mixture representation of the system reliability based
on the notion of the signature of the system. Some examples of weighted k-out-of-n
systems are presented to demonstrate the proposed models numerically and graphi-
cally.
Keywords: Weighted k-out-of-nsystems, Signature vector, Preventive mainte-
nance, System availability, Cost criterion.
∗Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81744,
Iran, (e-mail: k.hamdan@sci.ui.ac.ir)
†Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81744,
Iran, (e-mail: m.tavangar@sci.ui.ac.ir)
‡Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, 81744,
& School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746,
Tehran, Iran, e-mail: m.asadi@sci.ui.ac.ir
1
1 Introduction
1.1 Motivation and related literature
The class of weighted k-out-of-nsystems is an important class among the class of n-
component structures built up of redundant components. Because of the wide range of
applications, different stochastic and reliability properties of these systems have been in-
vestigated in the past three decades. In a weighted k-out-of-nsystem, each component
carries a positive integer weight which refers to the load or capacity of the component.
Consider the vector w= (w1, w2, ..., wn), where widenotes the weight of ith component in
the system, i= 1,2, ..., n. A weighted k-out-of-nsystem works if and only if the accumu-
lated weights of the working components is at least k(or, equivalently, the accumulated
weights of the failed components is at most Pn
i=1 wi−k), where kis a predetermined
threshold that satisfies min{w1, w2, ..., wn} ≤ k≤Pn
i=1 wi. If w1=w2=· · · =wn= 1,
the system reduces to ordinary k-out-of-nsystem that works if and only if at least k
components out of ncomponents work. An example of a weighted system is heating, ven-
tilation, and air conditioning (HVAC) system, in which the heating (cooling) units can be
considered as components with different capacities (wights). The failure of units leads to
a reduction of the system capacity and hence may cause the failure of the whole system.
A lighting system with a number of light bulbs of different wattage is another example
of weighted systems. For further examples of these systems, we refer to Samaniego and
Shaked (2008).
The notion of weighted k-out-of-nsystem was first proposed by Wu and Chen (1994),
where the authors proposed an algorithm for computing the reliability function of such
systems. Then, the stochastic and reliability properties of the weighted k-out-of-nsys-
tems have been investigated in the literature under different scenarios. For example, the
multi-state weighted k-out-of-nsystems and their reliability evaluation are studied by Li
and Zuo (2008). Eryilmaz (2013) studied the case that the components of the system
2
have random weights. The reliability properties of a weighted system consisting of two
different types of components are investigated by Eryilmaz and Sarikaya (2014). The per-
formance of weighted k-out-of-nsystems with dependent component lifetimes is explored
by Li et al. (2016). Rahmani et al. (2016) studied the stochastic ordering properties
in the class of weighted k-out-of-nsystems. The optimal allocation of active redundant
components in weighted k-out-of-nsystems is considered by Zhang (2018). Zhang et
al. (2018) investigated the total capacity of k-out-of-nsystems with random weights. The
reliability modeling of weighted k-out-of-nsystems with randomly chosen components is
investigated by Salehi et al. (2018). Eryilmaz and Bozbulut (2019) studied the case that
a weighted k-out-of-nsystem consists of components with three states. The reliability
analysis of weighted k-out-of-nsystems with heterogeneous components having random
weights is investigated by Zhang (2020).
In real-life situations, the systems are always at risk of failure, and as time progresses
an operating system is more affected by internal or external shocks. The sudden failure
of some systems, such as the failure of an aircraft, a bridge, or a nuclear system, not
only is very costly but also may be very dangerous. Hence, by taking into account
the cost of maintenance activities and other limitations, it is essential (or even vital)
to repair the operating system before the failure takes place. In other words, it is very
important to perform the “preventive maintenance” to reduce the likelihood of failures
of such systems. Several researchers have discussed different maintenance strategies for
coherent systems and various maintenance models have been proposed both for single-
and multi-unit systems.
Maintenance models mainly fall into two major classes: corrective maintenance (CM)
and preventive maintenance (PM). The CM is a repair action that is performed whenever
the system fails. According to MIL-STD-721B, a CM means to perform all actions, as
a result of a failure, to restore a system to a specified condition. Simply, the CM action
may be done by replacing the failed system with a new one or just remove the failure
3
by repairing the system and bring back it to the working state. The PM is an action
performed to a system which is still operating but may function in unsatisfactory working
conditions. According to MIL-STD-721B, PM means “all actions performed in an attempt
to retain a system in specified conditions by providing systematic inspection, detection,
and prevention of incipient failures”.
Among the various maintenance and replacement policies, age replacement policy might
be the most common and popular maintenance policy. According to this policy, the PM
or the CM actions for a system is always performed at its age TP M or at the failure of
the system, whichever occurs first. In the context of reliability, Barlow and Hunter (1960)
first introduced the concepts of minimal repair and especially imperfect maintenance. Af-
terwards, several maintenance models and modifications of the age replacement policy
were proposed for system of components. For example, Nakagawa (1984) proposed a new
minimal repair policy in which the system is replaced at time tor at the time that the kth
failure occurs, whichever occurs first. Various optimal policies on the imperfect mainte-
nance are discussed and summarized in Pham and Wang (1996). For further references, we
refer, among others, to Gertsbakh (2000), Nakagawa (2005), Levitin et al. (2018), Safaei
et al. (2018), Bad´ıa et al. (2018), Hashemi and Asadi (2020) and Bad´ıa et al. (2020).
In last two decades, attempts have been also made to introduce and apply new main-
tenance scheduling to systems consisting of several components. Wang and Pham (2006)
investigated the reliability and maintenance strategies for the multi-component systems.
Finkelstein and Gertsbakh (2015, 2016) studied preventive maintenance models for sys-
tem (network) where the component failures are modeled based on shock models. Cha et
al. (2017) considered age replacement of components operating in a random environment
modeled by a Poisson process of shocks. George-Williams and Patelli (2017) identified
the optimal maintenance strategy for a multi-state system and studied the problem of
identifying optimum number of maintenance teams employed and assigning them to com-
ponents. Recently, Hashemi and Asadi (2020) proposed some maintenance policies for a
4
general coherent system. To further study on system’s maintenance strategies, we refer
the reader, among many others, to Nicolai and Dekker (2008), Eryilmaz (2018), Mullor
et al. (2019), Zarezadeh and Asadi (2019), Zhang et al. (2020), Wang et al. (2020), Vu
et al. (2020), Eryilmaz (2020), Castro et al. (2020) and Shi et al. (2020).
According to the best of our knowledge, a few maintenance models have been introduced
for the weighted k-out-of-nsystems despite a wide range of their applications in various
areas in industrial and systems engineering. Khorshidi et al. (2016) considered a dynamic
evaluation of the multi-state weighted k-out-of-nsystem. The authors used the expected
failure cost of components as an unreliability index to compare the system unreliability
and system cost in order to make decision. The transition probabilities through the op-
erating states over time follow exponential distribution and the dynamic behavior of the
system is assessed by recursive algorithm approach. Atashgar and Abdollahzadeh (2017)
proposed a model to optimize at the same time the redundancy and imperfect opportunis-
tic maintenance of multi-state weighted k-out-of-nsystems. It is assumed by the authors
that the status of the components degrade over time and a condition-based opportunis-
tic maintenance approach based on different thresholds for a component health state is
developed.
1.2 Main contributions of the paper
In the present paper, we introduce some new age-based preventive maintenance strategies
for weighted k-out-of-nsystems. The main contributions of the present paper are as
follows. We assume that a weighted k-out-of-nsystem starts working at time t= 0. When
the system is scheduled to be repaired, we assume that there exist some components in the
system that have failed and the others are still working. As the functioning components
may have poor conditions, they are less resistant against shocks, and hence they need a
check or some lubrication, etc. In our proposed optimal maintenance policy, we perform
PM action on the unfailed but deteriorating components together with a CM action on
5
the failed components either at the time when the total weight of the working components
becomes less than or equal to a predetermined threshold m, where k≤m≤Pn
i=1 wi,
or at the age TP M , whichever occurs first. Under the assumption that all maintenance
actions take negligible times, we formulate the expected system maintenance cost per
unit of time using the signature based representation of the system reliability. Then, we
aim to minimize the proposed cost function as a target function of TP M , to obtain the
optimum value of TP M . We also maximize the long-run availability of the system, under
the conditions that the maintenance times are not negligible, in terms of the decision
variable TPM .
1.3 Organization of the paper
The remaining of the paper is organized as follows. In Section 2, after introducing the
necessary notations and abbreviations, we describe the system, represent the proposed
maintenance strategy, and determine the decision variables. Also, in this section, we
formulate our proposed cost function by calculating the expected costs of the failed com-
ponents at the time of system failure or at the time of PM action. In order to write the
formula for the cost function, we utilize the mixture representation of the system reliability
based on the notion of signature (cf. Samaniego (1985) and Franko and T¨ut¨unc¨u (2015)).
We then consider the long-run availability of the weighted k-out-of-nsystems and derive
a formula for the availability criterion on using the signature-based representation of the
system reliability. Section 3 is devoted to three examples of weighted k-out-of-nsystems,
to illustrate the proposed PM model numerically and graphically. The paper is finalized
by some concluding remarks in Section 4.
2 Description of the strategy
Before giving the description of our strategy, we introduce the following notations and
abbreviations that will be used throughout the paper.
6
CM corrective maintenance
PM preventive maintenance
X1, X2, ..., Xncomponent lifetimes
X1:n, X2:n, ..., Xn:nordered component lifetimes
Fi:n(t) distribution function of Xi:n
TP M preventive maintenance time
Tm,n time at which the accumulated weights of
working components is less than or equal to m
NTP M number of the failed components at TP M
NTm,n number of the failed components at Tm,n
piith element of signature vector
c1cost of CM on each component
c2cost of PM on each component
εi(t) the state of ith component at time t
wiweight of the ith component in the system
tmtime to perform CM
tTP M time to perform PM
Consider a binary-state (two-state) weighted k-out-of-nsystem consisting of ncom-
ponents with the weights vector w= (w1, w2, ..., wn), where wiis the weight of the ith
component. Let ε(t)=(ε1(t), ε2(t), ..., εn(t)) be the vector of components states, where
εi(t) = 1 if the ith component is working at time tand εi(t) = 0, otherwise. The weighted
k-out-of-nsystem works at a time instant tif the accumulated weight of the working
components at tis greater than or equal to a constant threshold k. In other words, if
φ(ε(t)) denotes the system state at t, then
φ(ε(t)) =
1,if Pn
i=1 wiI{εi(t)=1}≥k,
0,otherwise,
where I{·} denotes the indicator function.
7
Let X1, X2, ..., Xnbe nindependent and identically distributed (i.i.d) random variables
denoting the component lifetimes with an absolutely continuous cumulative distribution
function F. Suppose that X1:n, X2:n, ..., Xn:nare the ordered lifetimes of components.
Using the systems signature vector, one can easily obtain the reliability function of the
system in terms of the reliability functions of the Xi:n,i= 1, . . . , n, where computing
the signatures depends strongly on the weights of the components. We will show this in
Equation (3) in below.
Assume that the system starts operating at time t= 0. We introduce our maintenance
policy as follow: performing PM on the unfailed but deteriorating components together
with a CM on the failed components either at the time when the total weights of the
working components becomes less than or equal to a predetermined threshold m, where
k≤m≤Pn
i=1 wi, or at the age TP M , whichever occurs first. In other words, if at any
time instant in the time interval (0, TP M ), the accumulated weight of the functioning
components is less than or equal to the threshold value m, then the operator performs
a maintenance action. Otherwise, it is obvious that at the time instant TPM , the total
weight of the working components is greater than m, i.e., the system is still working at
time TP M . However, though the system is still working at TP M , to prevent the failure of
the system it is recommended to perform a PM at time TP M .
It is worth mentioning here that in the special case when m=k, the strategy becomes
to perform the preventive maintenance at the time of the system failure or at time TP M ,
whichever occurs first. In this case, the policy coincides with the classical PM policy.
Cost criterion
In the following, we aim to determine the decision parameter TP M that minimizes the
cost function. In the policy described above, at the time of PM, the failed components
are replaced by new components from the same population at a cost c1and the unfailed
components are perfectly repaired at a cost c2. Typically, c1c2. We first assume that
8
all maintenance actions take negligible times. The expected system maintenance cost per
unit time is then given by
ζ(TP M ) = C(TP M )
D(TP M ),(1)
where C(TP M ) is the expected system maintenance cost per renewal cycle and D(TP M )
is the expected duration of a renewal cycle.
Let Tm,n represent the time instant at which the accumulated weights of working com-
ponents of a weighted k-out-of-nsystem will be less than or equal to m. Note that Tm,n
is, in fact, the lifetime of a weighted m-out-of-nsystem. Let also NTP M and NTm,n denote
the number of the failed components at the time instants TPM and Tm,n, respectively.
Note that, according to the proposed policy, if the total weight of the working components
reaches the threshold m, then there are NTm,n failed components which are replaced. The
remaining (n−NTm,n ) components are functioning yet and the operator perform a perfect
PM on them. These actions lead to the cost c1NTm,n for replacing the failed components
and the cost c2(n−NTm,n ) for repairing the remaining ones. Therefore, the expected
system maintenance cost per renewal cycle can be written as
C(TP M ) = Ec1NTm,n + (n−NTm,n )c2I{Tm,n≤TP M }
+ (c1NTP M + (n−NTP M )c2)I{Tm,n>TP M }.
In order to evaluate C(TPM ), we need to compute the following two expectations
E1≡E{NTm,n I{Tm,n≤TP M }}
and
E2≡E{NTP M I{Tm,n >TP M }}.
The computation of E1and E2relies on the concept of the signature vector, which is
a useful tool to assess the reliability of the lifetimes of coherent systems and to make
comparisons between the systems based on various stochastic orders (see, Samaniego,
1985). Franko and T¨ut¨unc¨u (2015) generalized a variant of the notion of the signature
9
for weighted m-out-of-nsystems, and proposed an algorithm for calculating the signature
vector of such systems. They showed that the reliability function of the system can be
presented as a mixture of the reliability functions of the ordered component lifetimes,
based on the signature vector, as
R(t) = P(Tm,n > t) =
n
X
i=1
piP(Xi:n> t),(2)
where Xi:nrepresents the ith ordered component lifetime and piis the ith element of
the weighted signature vector of the system. piis, in fact, the probability that the
ith component failure leads to the system breakdown; i.e., pi=P(Tm,n =Xi:n). The
signature pican be easily computed as (see Franko and T¨ut¨unc¨u (2015))
pi=rn−i+1(n)
n
n−i+1−rn−i(n)
n
n−i, i = 1,2, ..., n, (3)
where ri(n), i= 1,2, ..., n, denotes the number of path sets including precisely iworking
components in the system. In other words, for the weighted-m-out-of-nsystem, ri(n)
refers to the number of sets of components that includes exactly icomponents for which
the sum of their weights is greater than or equal to the threshold m.
Based on the discussion given above, one can show that E1is given as follows:
E1=
n
X
i=1
ipiFi:n(TP M ),(4)
where Fi:n(t) = P(Xi:n≤t) and piis as defined in (3), i= 1, . . . , n. Similarly,
E2=
n
X
i=1
in
i¯
PiFi(TP M )¯
Fn−i(TP M ),(5)
where ¯
F(t)=1−F(t) is the common reliability function of the components lifetime and
¯
Pi=Pn
j=i+1 pj,i= 1, . . . , n (see, the Appendix for the proofs).
On the other hand, the expected length of a cycle is expressed as
E{min(Tm,n, TP M )}=ZTP M
0
¯
FTm,n (t)dt,
where ¯
FTm,n (t) is the reliability function of a weighted m-out-of-nsystem at the time
instant t. Using Equation (2), the expected length of one cycle can be represented as
E{min(Tm,n, TP M )}=
n
X
i=1
piZTP M
0
¯
Fi:n(t)dt, (6)
10
where ¯
Fi:n(t)=1−Fi:n(t). Hence, Equations (4), (5) and (6), imply that the cost function
ζ(TP M ) can be expressed as
ζ(TP M ) = c1E1+nc2P(Tm,n ≤TP M )−c2E1+c1E2+nc2P(Tm,n > TP M )−c2E2
E{min(Tm,n, TP M )}
=(c1−c2)(Pn
i=1 ipiFi:n(TP M ) + Pn
i=1 in
i¯
PiFi(TP M )¯
Fn−i(TP M )) + nc2
Pn
i=1 piRTP M
0¯
Fi:n(t)dt .(7)
Our aim here is to minimize the cost function ζ(TP M ) with respect to the decision variable
TP M .
An important question that arises here is whether there exists a finite optimal value for
the cost function ζ(TP M ). In the following proposition, we provide sufficient conditions for
the existence of the optimal value T∗
P M minimizing ζ(TP M ). Before that, let h(t) = f(t)
¯
F(t),
be the common failure rate function of the components, where f(t) denotes the density
function corresponding to F(t).
Proposition 2.1 Let h(t)be the common failure rate function of the components and
ζ(TP M )be as given in Equation (7). Assume that n0= max{i:pi>0}.If
lim
t→∞ h(t)>nc2
(n−n0+ 1)(c1−c2)µS
,(8)
then there exist a finite T∗
P M which minimizes ζ(TP M ), where from (2), µS=Pn0
i=1 piµi:n,
and µSand µi:ndenote the mean lifetime of the whole system and the mean of the ith
order statistic Xi:n, respectively.
Proof. See the Appendix.
Availability criterion
Some systems need very high requirements for the availability and reliability, and it is
extremely important to achieve the most time of availability for such systems regardless
of the cost of the maintenance activities. In this case, again we consider the system to be
served either at the time when the total weights of working components become less than
or equal to a predetermined threshold m, i.e., at Tm,n, or when the lifetime of the system
11
reaches TP M , whichever occurs first. It is supposed in the previous strategy that replacing
the failed component and repairing the unfailed one took a negligible time. However, we
assume that when the system is repaired at Tm,n, that is, when Tm,n ≤TP M , this will
last tmunit of time and in the other case, i.e., when Tm,n > TP M , this will take tTP M unit
of time. Typically, at the time Tm,n the total lost weights in the system is much more
than the total lost weights at time TPM , and as a result, we assume that the system will
take more time to be served, i.e., tm> tTP M . The goal of this strategy is to maximize
the availability of the system. The reward in this strategy is the total operational ‘up’
time of the system. Under these conditions, one renewal period will be equal to either
S=Tm,n +tmor S=TP M +tTP M with probabilities Fs(TP M ) and ¯
Fs(TP M ), respectively,
where Fs(TP M ) = P(Tm,n ≤TP M ) and ¯
Fs(TP M )=1−Fs(TP M ). Thus, one renewal cycle
is equal to
S= min(Tm,n, TP M ) + tmI{Tm,n ≤TP M }+tTP M I{Tm,n>TP M },
with the mean duration
E(S) = E{min(Tm,n, TP M ) + tmI{Tm,n ≤TP M }+tTP M I{Tm,n>TP M }}(9)
=ZTP M
0
¯
Fs(t)dt +tm(1 −¯
Fs(TP M )) + tTP M ¯
Fs(TP M ).
The mean reward per unit of time is the system availability:
η(TP M ) = RTP M
0¯
Fs(t)dt
RTP M
0¯
Fs(t)dt +tm(1 −¯
Fs(TP M )) + tTP M ¯
Fs(TP M ).
Using the weighted signature vector p= (p1, p2, ..., pn) and rewriting the last equation
using Equation (2), we obtain
η(TP M ) = RTP M
0Pn
i=1 pi¯
Fi:n(t)dt
RTP M
0Pn
i=1 pi¯
Fi:n(t)dt +tm(1 −Pn
i=1 pi¯
Fi:n(TP M )) + tTP M Pn
i=1 pi¯
Fi:n(TP M ).
We are interested in maximizing the availability function with respect to the decision
variable TPM .
Remark 2.2 To explore the existence of a finite optimal value T∗
P M for the availability
η(TP M ), assume again that h(t) is the common failure rate function of the components
12
and n0= max{i:pi>0}. It can be shown, using a similar argument as used to prove
Proposition 2.1, that a sufficient condition for the existence of the optimal value T∗
P M
maximizing η(TP M ) is that
lim
t→∞ h(t)>tm
(n−n0+ 1)(tm−tP M )µS
,
where µSdenotes the mean lifetime of the whole system.
3 Illustrative Examples
In this section, we provide some examples to examine the theoretical results of previous
sections.
Example 3.1 Consider a Gas pipeline network configuration consisting of some pipes
and 5 compressor stations. The compressor stations, also called pumping stations, are
located in transmission lines and transport Gas from one location to another one. Usually,
they may run below their nominal capacity. Suppose that the compressor stations have
different capacities w1= 1, w2= 2, w3= 3, w4= 4 and w5= 6 which can be considered
as the weights of pumping stations. Hence, the vector of the weights of the system’s
components is w= (1,2,3,4,6). It is assumed that the system will fail when the total
weight (capacity) of the working components (compressor stations) is less than or equal
to k= 8; i.e., the system has a weighted 8-out-of-5 structure. If we take m= 9, then the
system will be preventively repaired when the accumulated weights of working components
is less than or equal to 9. Using Equation (3),the signature vector is computed, in Franko
and T¨ut¨unc¨u (2015), as p= (0,0.3,0.5,0.2,0). The cost of replacing a failed compressor
is estimated as c1= 150000$ where the preventive maintenance of an unfailed compressor
costs c2= 6000$. It is also assumed that the lifetime of the compressors follow the Weibull
distribution W(α, β) with density function
f(x) = αβxα−1e−β xα, x > 0.
13
The cost function (7) of the proposed maintenance policy as a function of TP M are depicted
for different values of c1and c2in Figure 1.
c1=8
c1=12
c1=18
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
10
20
30
40
50
TPM
ΖHTPML
(a)
c2=2.0
c2=1.2
c2=0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
10
20
30
40
50
TPM
ΖHTPML
(b)
Figure 1: The plots for the cost function in Example 3.1; (a) c2= 0.6, (b) c1= 15.
In Figure 1(a), the cost functions of the strategy for a fixed value of c2and different
values of c1are plotted. In Figure 1(b), a similar graph is depicted for different values
of c2and a fixed value of c1. The orange dots in the graphs are the minimum cost for
various combinations of c1and c2. It is observed that when the cost of CM gets bigger,
then the optimal time of PM decreases and, as expected, the optimum value of the cost
function increases. Also, as the cost of PM increases, then so do both the optimal time
of PM and the optimal cost function.
Now let us assume that the components lifetimes are distributed as a gamma distribu-
tion Γ(α, β) with density function
f(x) = 1
Γ(α)βαxα−1e−x/β, x > 0.
The plots of cost functions for different values of c1and c2are presented in Figure 2.
The interpretation is similar to Figure 1(b). The same results are confirmed, for different
values of c1and c2, by the numerical optimized values for TP M in Table 1.
14
c2=4.5
c2=2.3
c2=0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
10
20
30
40
50
TPM
ΖHTPML
Figure 2: The cost functions in Example 3.1 for the gamma distribution with c1= 25.
Lifetime Distribution c1c2T∗
P M ζ(T∗
P M )
W(3,2)
15
2 0.787 17.42
1.2 0.716 12.707
0.5 0.520 7.237
18
0.6
0.520 8.684
12 0.601 7.531
8 0.698 6.506
Γ(3,2) 25
4.5 1.635 28.285
2.3 0.982 20.643
0.8 0.577 11.317
Table 1: Optimized cost function in Example 3.1
Assume again that the lifetime of the compressors follow the Weibull W(3,2) distribu-
tion. The availability of the system is plotted in Figure 3(a), for tm= 3.4 and different
values of tTPM , and in Figure 3(b) for tTP M = 0.4 and different values of tm. The orange
dots in the graphs are the maximum availability towards each combination of c1and c2.
The numerical values for the optimized time TP M for different values of tmand tTPM are
computed in Table 2. As it is seen, when tTP M increases, then so does TP M and the time
of PM is postponed. In this case, the availability decelerates. Also one can observe that
15
the larger tmresults in the smaller both TP M , and the system availability. This makes
sense since with increasing tm, the accumulated weight of the working components should
not reach mand this, in turn, makes to avoid wasting too much time tm.
tTPM =1.3
tTPM =1.0
tTPM =0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TPM
ΗHTPML
(a)
tm=6
tm=3.8
tm=2.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TPM
ΗHTPML
(b)
Figure 3: The plots of the availability function in Example 3.1 for Weibull distribution; (a)
tm= 3.4, (b) tTP M = 0.4.
Lifetime Distribution tmtTP M T∗
P M η(T∗
P M )
W(3,2)
3.4
1.3 1.22 0.415
1.0 1.08 0.492
0.6 0.98 0.586
6.0
0.4
0.77 0.62
3.8 0.95 0.67
2.6 1.09 0.69
Table 2: Optimized availability in Example 3.1
In the next example, we investigate the optimum value of TPM in the case that the
threshold mis predetermined such that the system undergoes PM before the system
failure takes place.
Example 3.2 Consider a 6-component weighted system with component weights w=
(3,4,4,5,6,13). Suppose that the system works when the total weights of operating
components is at least equal k= 10. The cost of replacing the failed component and
16
the cost of partial repairing of the unfailed component are estimated as c1= 45 and
c2= 15, respectively. The component lifetimes are assumed to be i.i.d random variables
with Weibull distribution W(5,2). This system is going to be served either when the
accumulated weights become less than or equal to m= 12, or at the time TP M , whichever
occurs first. In this case the system will never fail and maintenance activities will be
performed before the failure takes place. For the case of m= 12, we have shown that
the vector of signature is p= (0,0,0.05,0.616,0.16,0.16). The cost function for different
values of costs c1and c2is plotted in Figures 4(a) and 4(b). The numerical computations
are given in Table 3. Again, an increase in c1will result in a decrease in T∗
P M , whereas
with increasing in c2,T∗
P M increases too.
c1=150
c1=100
c1=50
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
50
100
150
200
250
300
TPM
ΖHTPML
(a)
c2=3.8
c2=1.6
c2=0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
20
40
60
80
100
120
TPM
ΖHTPML
(b)
Figure 4: The plots of the cost function in Example 3.2: (a) c2= 15, (b) c1= 45.
Lifetime Distribution c1c2T∗
P M ζ(T∗
P M )
W(5,2)
45
3.8 0.946 30.215
1.6 0.784 15.305
0.8 0.680 8.826
150
15
0.983 114.855
100 1.082 104.525
50 1.315 86.915
Table 3: Optimized cost function in Example 3.2
17
The availability of the system consisting of components with lifetimes having Weibull
W(5,2) distribution are depicted in Figures 5(a) and 5(b) for different values of tmand
tTP M . Similar graphs, with component lifetimes having gamma Γ(3,1) distribution, are
plotted in Figures 6(a) and 6(b). The numerical values of the optimized T∗
P M are also
computed in Table 4.
tTPM =1.5
tTPM =1.0
tTPM =0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TPM
ΗHTPML
(a)
tm=15
tm=10
tm=6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TPM
ΗHTPML
(b)
Figure 5: The plots of the availability function in Example 3.2 for Weibull distribution (a)
tm= 10.5, (b) tTP M = 0.8.
tTPM =1.5
tTPM =1.0
tTPM =0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TPM
ΗHTPML
(a)
tm=15
tm=10
tm=6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TPM
ΗHTPML
(b)
Figure 6: The plots of the availability function in Example 3.2 for the gamma distribution: (a)
tm= 10.5, (b) tTP M = 0.8.
In the proposed strategy, we have assumed so far that mis a predetermined threshold
ranges between k≤m≤Pn
i=1 wi. In the following example, we provide an example to
show the situation that the time of PM, TP M , is assumed to be predetermined, and the
18
Lifetime Distribution tmtTP M T∗
P M η(T∗
P M )
W(3,2)
10.5
1.5 1.03 0.409
1.0 1.18 0.502
0.6 1.25 0.617
15
0.8
1.11 0.72
10 1.19 0.83
6.0 1.27 0.91
Γ(3,1)
10.5
1.5 1.71 0.47
1.0 1.55 0.56
0.6 1.38 0.67
15
0.8
1.42 0.58
10 1.65 0.62
6.0 1.74 0.65
Table 4: Optimized availability in Example 3.2
decision variable of interest under which the cost function is optimized is the threshold
m.
Example 3.3 Consider a 15-component weighted system with component weights
w= (2,2,3,4,4,5,6,6,6,6,8,9,9,9,10)
and the i.i.d. component lifetimes distributed as Weibull distribution W(5,2). Let the
system function if the total weights of operating components is at least equal to k= 62.
Assume that the threshold mis the decision variable and the PM time is a predetermined
constant, say, TP M = 1.3. The plot of the cost function ζas a function of mis depicted
in Figure 7(a) for c1= 100 and c2= 15. It is evident that the cost function ζhas a
minimum at m∗= 79 with the optimum cost 278.203. More numerical computations are
given in Table 5. It is seen that an increase in c1will result in an increase in m∗, while
with increasing in c2,m∗decreases.
19
50 60 70 80
278
279
280
281
m
ζ
(a)
65 70 75 80 85
0.680
0.685
0.690
0.695
0.700
0.705
0.710
0.715
m
η
(b)
Figure 7: The plots of the cost and availability as functions of min Example 3.3.
Lifetime Distribution c1c2m∗ζ∗
W(5,2)
100
1 88 92.131
15 79 278.203
30 2 434.637
150
15
88 320.652
100 79 278.203
50 2 217.319
Table 5: Optimized cost function in Example 3.3
To investigate the availability of the system under the proposed strategy, we assume
that the PM time is a fixed constant, say, TPM = 1.6. First, note that we have assumed
so far that tm> tTP M . In this case, our numerical computations show that the availability
is a decreasing function of mand hence the optimum value m∗maximizing the system
availability is m∗=k= 62. This situation has an intuitive reason, since as tm> tTP M ,
the value of mshould be as small as possible so that Tm,n be as large as possible in
expectation. On the other hand, if for any reason one can assume that tm≤tTPM , we
show in this example that the system availability does not necessarily take its maximum
at m=k. For example, if tm= 0.5 and tTP M = 0.8, then the numerical calculations show
that the optimum value of mis 77 with the optimum availability 0.717; see Figure 7(b).
20
More numerical computations are given in Table 6. It is observed that an increase in tm
will result in a decrease in m∗, whereas with increasing in tTP M ,m∗increases too.
Lifetime Distribution tmtTP M m∗η∗
W(5,2)
0.5
0.6 68 0.735
0.8 77 0.717
1 79 0.709
0.2
0.8
83 0.844
0.4 79 0.753
0.6 71 0.690
Table 6: Optimized availability function in Example 3.3
4 Conclusions
In the present paper, we introduced some new optimal preventive maintenance strategies
for weighted k-out-of-nsystems. Assuming that a weighted k-out-of-nsystem starts
operating at time t= 0, the following maintenance policy for the system is proposed. We
performed preventive maintenance action on the unfailed but deteriorating components
together with a corrective maintenance action on the failed components either in the case
that the total weights of the working components is less than or equal to a threshold m,
where k≤m≤Pn
i=1 wi, or when the system attains the age TP M , whichever occurs first.
Using the signature-based mixture representation of the system reliability, we formulated
the expected system maintenance cost per unit of time to obtain the optimum value of
preventive maintenance time. We also formulated the long-run availability of the system,
under the conditions that the maintenance times are not negligible. Several illustrative
examples are presented to interpret the theoretical results of the paper numerically and
graphically. Throughout the paper, we assumed that the decision variable of interest,
under which the cost (availability) function is optimized, is the time of PM action and
21
the threshold mis a fixed value. We also gave an example of a 15-component system
for which the cost function of the proposed strategy is optimized, in the case that the
threshold mis considered as the decision variable and the time of PM is assumed to be a
fixed value. Considering the problem of optimization of the cost and availability functions
in the case the both quantities TP M and mare jointly considered as the decision variables
might be an interesting problem.
Although under the settings described in the paper, our proposed optimal policies pro-
vides the feasibility of maintaining the weighted k-out-of-nsystems in optimal conditions,
there are still some limitations on employing the proposed criteria. Throughout the pa-
per, we assumed that the cost of repairing the failed (and unfailed) components is the
same for all components, irrespective of the weights of the components. It would be an
appealing problem to investigate the optimal maintenance of the system under the condi-
tion that the costs of repairing (renewing) the components are functions of the weights of
the components. Another interesting problem that can be verified, as a future study, is to
consider the case that the components of the system are non-identical and/or statistically
dependent.
22
Appendix
Proofs of Equations (4) and (5)
We have
E1=E{NTm,n I{Tm,n≤TP M }}
=
n
X
i=1
iP (NTm,n I{Tm,n≤TP M }=i)
=
n
X
i=1
i
n
X
j=1
P(NTm,n I{Tm,n≤TP M }=i|Tm,n =Xj:n)P(Tm,n =Xj:n)
=
n
X
i=1
i
n
X
j=1
P(NTm,n =i, Tm,n ≤TP M |Tm,n =Xj:n)pj
=
n
X
i=1
i
n
X
j=1
P(Tm,n =Xi:n, Xj:n≤TP M |Tm,n =Xj:n)pj
=
n
X
i=1
ipiFi:n(TP M ),
where ¯
F(t) = 1 −F(t) is the common reliability function of the components lifetime and
piis as defined in (3), i= 1, . . . , n. Similarly,
E2=E{NTP M I{Tm,n >TP M }}
=
n
X
i=1
iP (NTP M I{Tm,n >TP M }=i)
=
n
X
i=1
i
n
X
j=1
P(NTP M I{Tm,n >TP M }=i|Tm,n =Xj:n)P(Tm,n =Xj:n)
=
n
X
i=1
i
n
X
j=1
P(NTP M =i, Tm,n > TP M |Tm,n =Xj:n)P(Tm,n =Xj:n)
=
n
X
i=1
i
n
X
j=1
P(Xi:n≤TP M < Xi+1:n, Xj:n> TP M |Tm,n =Xj:n)pj
=
n
X
i=1
i
n
X
j=i+1
pjP(Xi:n≤TP M < Xi+1:n)
=
n
X
i=1
in
iFi(TP M )¯
Fn−i(TP M )
n
X
j=i+1
pj
=
n
X
i=1
in
i¯
PiFi(TP M )¯
Fn−i(TP M ),
where ¯
Pi=Pn
j=i+1 pj,i= 1, . . . , n.
23
Proof of Proposition 2.1
First note that the cost function (7) can be written as
ζ(TP M ) = (c1−c2)(Pn
i=1 ipi¯
Fi:n(TP M ) + Pn−1
i=1 i¯
Pi[Fi:n(TP M )−Fi+1:n(TP M )] + nc2
Pn
i=1 piRTP M
0¯
Fi:n(t)dt .
On differentiating ζ(TP M ) with respect to TP M , we obtain
dζ(TP M )
dTP M
sgn
=ψ(TP M ),(10)
where
ψ(TP M ) = Pn
i=1 ipifi:n(TP M ) + Pn
i=1 i¯
Pi[fi:n(TP M )−fi+1:n(TP M )]
Pn
i=1 pi¯
Fi:n(TP M )
−(c1−c2)(Pn
i=1 ipi¯
Fi:n(TP M ) + Pn
i=1 i¯
Pi[Fi:n(TP M )−Fi+1:n(TP M )]) + nc2
(c1−c2)Pn
i=1 piRTP M
0¯
Fi:n(t)dt ,
and ‘sgn
= ’ means to have the same sign. It can be shown, after some manipulations, that
lim
TP M →∞ ψ(TP M )=(n−n0+ 1) lim
TP M →∞ h(TP M )−nc2
(c1−c2)Pn0
i=1 piµi:n
.
Hence, with the stated assumption in the proposition, we have limTPM →∞ ψ(TP M )>0.
By continuity of ψ, there exists a t0such that ψ(t)≥0 for t>t0.This means that ζ(TP M )
is eventually strictly increasing. On the other hand, one can obtain
lim
TP M →0ζ(TP M ) = +∞,
and thus ζ(TP M ) is initially decreasing. We conclude that ζ(TPM ) has at least a finite
minimum.
Acknowledgments:
We would like to express our sincere thanks to the associate editor and four anonymous
referees for their constructive comments and suggestions which improved the presentation
of the paper. Asadi’s research work was performed in IPM Isfahan branch and was in
part supported by a grant from IPM (No. 99620213).
24
References
[1] Atashgar, K. and Abdollahzadeh, H. (2017). A joint reliability and imperfect opportunistic
maintenance optimization for a multi-state weighted k-out-of-nsystem considering economic
dependence and periodic inspection. Quality and Reliability Engineering International,33(8),
1685–1707.
[2] Bad´ıa,F.G., Berrade, M.D., Cha, J.H. and Lee, H. (2018). Optimal replacement policy under a
general failure and repair model: Minimal versus worse than old repair. Reliability Engineering
& System Safety,180, 362–372.
[3] Bad´ıa, F.G., Berrade, M.D. and Lee, H. (2020). An study of cost effective maintenance policies:
Age replacement versus replacement after Nminimal repairs. Reliability Engineering & System
Safety,201, 106949.
[4] Barlow, R. and Hunter, L. (1960). Optimum preventive maintenance policies. Operations Re-
search,8(1), 90–100.
[5] Castro, I.T., Basten, R.J.I. and van Houtum, G-J. (2020). Maintenance cost evaluation for
heterogeneous complex systems under continuous monitoring. Reliability Engineering & System
Safety,200, 106745.
[6] Cha, J.H., Finkelstein, M. and Levitin, G. (2017). On preventive maintenance of systems with
lifetimes dependent on a random shock process. Reliability Engineering & System Safety,168,
90–97.
[7] Eryilmaz, S. (2013). On reliability analysis of a k-out-of-nsystem with components having
random weights. Reliability Engineering & System Safety,109, 41–44.
[8] Eryilmaz, S. (2018). The number of failed components in a k-out-of-nsystem consisting of
multiple types of components. Reliability Engineering & System Safety,175, 246–250.
[9] Eryilmaz, S. and Bozbulut, A.R. (2019). Reliability analysis of weighted k-out-of-nsystem
consisting of three-state components. Proceedings of the Institution of Mechanical Engineers,
Part O: Journal of Risk and Reliability,233(6), 972–977.
[10] Eryilmaz, S. and Sarikaya, K. (2014). Modeling and analysis of weighted k-out-of-n:Gsystem
consisting of two different types of components. Proceedings of the Institution of Mechanical
Engineers, Part O: Journal of Risk and Reliability,228(3), 265–271.
25
[11] Eryilmaz, S. (2020). Age-based preventive maintenance for coherent systems with applications
to consecutive-k-out-of-nand related systems. Reliability Engineering & System Safety,204,
107143.
[12] Finkelstein M. and Gertsbakh, I. (2015). ‘Time-free’ preventive maintenance of systems with
structures described by signatures. Applied Stochastic Models in Business and Industry,31(6),
836–845.
[13] Finkelstein M. and Gertsbakh, I. (2016). On preventive maintenance of systems subject to
shocks. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and
Reliability,230(2), 220–227.
[14] Franko, C. and T¨ut¨unc¨u, G.Y. (2015). Signature based reliability analysis of repairable
weighted k-out-of- n:Gsystems. IEEE Transactions on Reliability,65(2), 843–850.
[15] George-Williams, H. and Patelli, E. (2017). Maintenance strategy optimization for complex
power systems susceptible to maintenance delays and operational dynamics. IEEE Transac-
tions on Reliability,66(4), 1309–1330.
[16] Gertsbakh, I. (2000). Reliability theory: with applications to preventive maintenance. Springer,
Berlin.
[17] Hashemi, M. and Asadi, M. (2020). On the components failure in coherent systems with
applications to maintenance strategies. To appear in: Advances in Applied Probability.
[18] Hashemi, M., Asadi, M. and Zarezadeh, S. (2020). Optimal maintenance policies for coherent
systems with multi-type components. Reliability Engineering & System Safety,195, DOI:
10.1016/j.ress.2019.106674.
[19] Khorshidi, H.A., Gunawan, I. and Ibrahim, Y. (2016). A dynamic unreliability assessment
and optimal maintenance strategies for multistate weighted k-out-of-n:F.Applied Stochastic
Models in Business and Industry,32(4), 485–493.
[20] Levitin, G., Finkelstein, M. and Dai, Y. (2018). Optimizing availability of heterogeneous
standby systems exposed to shocks. Reliability Engineering & System Safety,170, 137–145.
[21] Li, W. and Zuo, M.J. (2008). Reliability evaluation of multi-state weighted k-out-of-nsystems.
Reliability Engineering & System Safety,93(1), 160–167.
26
[22] Li, X., You, Y. and Fang, R. (2016). On weighted k-out-of-nsystems with statistically depen-
dent component lifetimes. Probability in the Engineering and Informational Sciences, 30(4),
533-546.
[23] MIL-STD-721B (1966). Definition of Effectiveness Terms for Reliability, Maintainability, Hu-
man Factors, and Safety. Department of Defence, Washington, D.C.
[24] Mullor, R., Mulero, J. and Trottini, M. (2019). A modelling approach to optimal imperfect
maintenance of repairable equipment with multiple failure modes. Computers & Industrial
Engineering,128, 24–31.
[25] Nakagawa, T. (1984). Optimal policy of continuous and discrete replacement with minimal
repair at failure. Naval Research Logistics Quarterly,31(4), 543–550.
[26] Nakagawa, T. (2005). Maintenance Theory of Reliability. Springer-Verlag, London.
[27] Nicolai, R. and Dekker, R. (2008). Optimal Maintenance of Multi-component Systems: A
Review. Complex System Maintenance Handbook, Springer, London, pp. 263–286.
[28] Pham, H. and Wang, H. (1996). Imperfect maintenance. European Journal of Operational
Research,94(3), 425-438.
[29] Rahmani, R. A., Izadi, M. and Khaledi, B. E. (2016). Stochastic comparisons of total capacity
of weighted k-out-of-nsystems. Statistics & Probability Letters,117, 216-220.
[30] Safaei, F., Ahmadi, J. and Gildeh, B.S. (2018). An optimal planned replacement time based
on availability and cost functions for a system subject to three types of failures. Computers &
Industrial Engineering,124, 77–87.
[31] Salehi, M., Shishebor, Z. and Asadi, M. (2019). On the reliability modeling of weighted
k-out-of-nsystems with randomly chosen components. Metrika,82(5), 589–605.
[32] Samaniego, F.J. (1985). On closure of the IFR class under formation of coherent systems.
IEEE Transactions on Reliability,34(1), 69–72.
[33] Samaniego, F.J. and Shaked, M. (2008). Systems with weighted components. Statistics and
Probability Letters,78, 815–823.
[34] Vu, H.C., Do, F., Fouladirad, M. and Grall, A. (2020). Dynamic opportunistic maintenance
planning for multi-component redundant systems with various types of opportunities. Relia-
bility Engineering & System Safety,198, 106854.
27
[35] Wang, H. (2002). A survey of maintenance policies of deteriorating systems. European Journal
of Operational Research,139, 469–489.
[36] Wang, H. and Pham, H. (2006). Reliability and Optimal Maintenance. Springer, London.
[37] Wang, Y., Liu, Y., Chen, J. and Li, X. (2020). Reliability and condition-based maintenance
modeling for systems operating under performance-based contracting. Computers & Industrial
Engineering,142, 106344.
[38] Wu, J. and Chen, R. (1994). An algorithm for computing the reliability of weighted k-out-of-n
systems. IEEE Transactions on Reliability,43(2), 327-328.
[39] Zarezadeh, S. and Asadi, M. (2019). Coherent systems subject to multiple shocks with appli-
cations to preventative maintenance. Reliability Engineering & System Safety, 185, 124-132.
[40] Zhang, Y. (2018). Optimal allocation of active redundancies in weighted k-out-of-nsystems.
Statistics & Probability Letters 135, 110-117.
[41] Zhang, Y., Ding, W. and Zhao, P. (2018). On total capacity of k-out-of-nsystems with
random weights. Naval Research Logistics,65(4), 347-359.
[42] Zhang, F., Shen, J. and Ma, Y. (2020). Optimal maintenance policy considering imperfect
repairs and non-constant probabilities of inspection errors. Reliability Engineering & System
Safety, doi: 10.1016/j.ress.2019.106615.
[43] Shi, Y., Zhu, W. , Xiang, Y. and Feng, Q. (2020). Condition-based maintenance optimization
for multi-component systems subject to a system reliability requirement. Reliability Engineer-
ing & System Safety,202, 107042.
[44] Zhang, Y. (2020). Reliability analysis of randomly weighted k-out-of-nsys-
tems with heterogeneous components. Reliability Engineering & System Safety,
doi.org/10.1016/j.ress.2020.107184.
28
