Uncertain Random Block Replacement Policy with No Replacement at
Failure
ZHENG WU
Aircraft Maintenance and Engineering Corporation Beijing
Beijing 100020
CHINA
XINWANG LI
College of Science
Civil Aviation University of China, CHINA
MIN FENG
College of Economics and Management
Civil Aviation University of China, CHINA
HEJIAO SHEN, CHUNXIAO ZHANG
College of Science
Civil Aviation University of China
Tianjin 300300
CHINA
Abstract: The classic block replacement problem assumes the lifetime of component as a random variable, and
the downtime cost caused by failed component is a constant. However, when a new type of unit is used in a block
replacement system, its lifetime is indeterminate and its distribution cannot be obtained by probability theory due
to the lack of historical operational data. And if the component is not replaced immediately when it fails, the
downtime cost per unit time is not fixed but always effected by some stochastic factors such as market. Thus,
uncertainty and randomness coexist in a block replacement system of new components. In this paper, the lifetime
of the component is regarded as an uncertain variable and the downtime cost per unit time is considered as a
random variable, and an uncertain random programming model of block replacement with no replacement at
failure is developed for a system composed of new type of components, and the formulated model aims to find
an optimal replacement time to minimize the expected cost rate in one replacement cycle. An implicit solution of
the optimal replacement time is obtained, and the condition ensuring the existence and uniqueness of the optimal
replacement time is given. Finally, a numerical example is presented to prove feasibility of the proposed model.
Key-Words: - Block replacement, No replacement at failure, Uncertainty theory, Uncertain random variable.
Received: January 19, 2023. Revised: July 21, 2023. Accepted: August 13, 2023. Published: September 20, 2023.
1 Introduction
Block replacement policy with no replacement at
failure means that a component is always replaced at
a fixed time, but not at failure. The policy is usually
applied to a complex maintenance system such as a
electronic system, where a component is not moni-
tored continuously and its failure can only be detected
at a fixed time. A key assumption in classic mod-
els of block replacement policy is that the lifetime of
component is a random variable, and the works were
based on probability theory adopting reliable histori-
cal data of the lifetimes. However, in most retail en-
vironments, especially for a new type of component,
there is no sample available to estimate the probabil-
ity distribution of lifetime , so we invite a number
of industry experts to give their subjective reliabil-
ity. Due to the tendency of humans towards events
that are unlikely to occur in, [1], the range of confi-
dence may be much larger than the actual frequency.
If we still use traditional random knowledge to deal
with such issues, there may be some errors with the
actual situation, [2]. To address the issues encoun-
tered above, [3], established the uncertainty theory in
2007, and was improved by, [4], in 2010 based on
normality, duality, subadditivity and product axioms.
Subsequently, many researchers made contributions
to the study of uncertainty theory.
At present, the uncertainty theory has been fur-
ther developed and promoted and has been applied in
many fields such as uncertain renewal process, [5],
uncertain calculus, [5], uncertain programming, [6],
uncertain statistics, [4], uncertain risk analysis, [7],
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uncertain reliability analysis, [7], uncertain inference,
[8], uncertain logic, [9], and uncertain finance, [10].
Based on the uncertain renewal process, [11], studied
the age substitution policy for random age. Assuming
the service lifetime of the component is an uncertain
variable, and finding the optimal time to replace the
component. [12], proposed a new uncertain age sub-
stitution policy, assuming the age of the component as
an uncertain variable, and discussed the optimal life
replacement strategy by treating the lifespan of parts
as different distribution uncertain variables.
Block replacement policy in uncertain environ-
ment was firstly proposed by, [13], where the com-
ponents are replaced at failure and fixed time and
the lifetimes of the components are assumed to be
independently identically distributed uncertain vari-
ables because of the absence of historical data. Un-
der the same assumption to the lifetime, [14], pro-
vided uncertain block replacement policy with no re-
placement at failure, and the downtime cost caused
by failed unit was considered as a constant in their
model. Actually, if the failed unit is not replaced im-
mediately, various kinds of stochastic factors have in-
fluence on the downtime cost, such as market, trans-
portation, the condition of management, and so on.
Thus, randomness and uncertainty may exist simulta-
neously in block replacement problem. [15], has pro-
posed the concepts of uncertain random variable and
chance theory which is a mathematical methodology
for modeling complex systems with not only uncer-
tainty but also randomness. As applications of chance
theory, uncertain random programming, [16], uncer-
tain random risk analysis, [17], uncertain random re-
liability analysis, [18], uncertain random graph, [19],
and uncertain random network, [19], have been de-
veloped successfully.
In this paper, we consider a system composed of
a new type of components and focus on studying a
block replacement strategy model with no replace-
ment at failure. Under the framework of chance the-
ory, the lifetime of the component will be assumed as
an uncertain variable, and the cost of downtime per
unit time from failure to replacement will be consid-
ered as a random variable, and an uncertain random
programming model will be developed to find an op-
timal replacement time by minimizing the expected
cost rate within a replacement cycle.
The rest of this article is organized as follows.
Section 2 provides some basic concepts and proper-
ties of uncertainty theory. In Section 3, the model of
block replacement policy with no replacement at fail-
ure based on chance theory is proposed and the op-
timal replacement policy is derived. A numerical ex-
ample is provided in Section 4. Some conclusions and
future prospects are made in Section 5. Finally, some
applications of this article are presented in Section 6.
2 Preliminaries
Let Γbe a nonempty set. A collection of Γis
called a σ-algebra. Uncertain measure is a function
from to [0,1]. To present an axiomatic definition of
uncertain measure, it is necessary to assign to each
event Λa number M{Λ}which indicates the belief
degree that the event Λwill occur. In order to en-
sure that the number M{Λ}has certain mathematical
properties, [3], proposed the following three axioms:
Axiom 1 (Normality Axiom) M{Γ}= 1 for the uni-
versal set Γ.
Axiom 2 (Duality Axiom) M{Λ}+M{Λc}= 1 for
any event Λ.
Axiom 3 (Subadditivity Axiom) For every countable
sequence of events Λ1,Λ2,··· we have
M∞
i=1
Λi≤
∞
i=1 M{Λi}.
Definition 1 (Liu [3]) Let ξbe a function from an
uncertain space (Γ,L,M)to a real set R, if for any
Borel set B, the set
{ξ∈B}={γ∈Γ|ξ(γ)∈B}
is an event.
Definition 2 (Liu [3]) Let ξis an uncertain variable,
then the function
Φ(x) = M{ξ≤x}
is called an uncertainty distribution of ξ.
Definition 3 (Liu [4]) Let ξbe an uncertain variable
with regular uncertainty distribution Φ(x). Then the
inverse function Φ−1(α)is called the inverse uncer-
tainty distribution of ξ.
Definition 4 (Liu [20]) For any Borel set
B1, B2,··· , Bmin the real number set R, if the
variable uncertain ξ1, ξ2,··· , ξmsatisfies
Mm
i=1
(ξi∈Bi)=
m
i=1 M{ξi∈Bi}
then they are called independent.
Theorem 1 (Liu [3]) Let ξbe an uncertain variable,
and its uncertainty distribution is Φ. If E[ξ]exists,
then
E[ξ] = +∞
0
(1 −Φ(x))dx −0
-∞
Φ(x)dx.
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Chance theory is a mathematical method used to
model complex systems containing both uncertain
and random variables. Let (Γ,L,M)be an uncer-
tainty space and let (Ω,A,Pr)be a probability space.
Then the product (Γ,L,M)×(Ω,A,Pr)is called a
chance space.
Definition 5 (Liu [15]) Let (Γ, , M)×(Ω,A,Pr)be
a chance space, and let Θ∈ L×A be an event. Then
the chance measure of Θis defined as
Ch {θ}=1
0
Pr {ω∈Ω|{γ∈Γ|(γ, ω)∈θ} ≥ r}dr.
Definition 6 (Liu [15]) An uncertain random vari-
able is a measurable function ξfrom a chance space
(Γ, , M)×(Ω,A,Pr)to the set of real numbers, i.e.,
ξ∈Bis an event for any Borel set B.
Theorem 2 (Liu [15]) Let f:Rn→ R be a mea-
surable function, and ξ1, ξ2,··· , ξnis an uncertain
random variables on the chance space (Γ, , M)×
(Ω,A,Pr). Then ξ=f(ξ1, ξ2,··· , ξn)is an uncer-
tain random variable determined by
ξ(γ, ω) = f(ξ1(γ, ω), ξ2(γ, ω),··· , ξn(γ, ω))
for any (γ, ω)∈Γ×Ω.
Theorem 3 (Liu [16]) Let η1, η2,··· , ηmbe in-
dependent random variables with probability dis-
tributions Ψ1,Ψ2,··· ,Ψm, and let τ1, τ2,··· , τn
be independent uncertain variables with uncer-
tainty distributions Υ1,Υ2,··· ,Υn, respectively.
If f(η1,··· , ηm, τ1,··· , τn)is a strictly increasing
function or a strictly decreasing function with regard
to τ1,··· , τn, then the uncertain random variable
ξ=f(η1,··· , ηm, τ1,··· , τn)
has an expected value
E[ξ] = Rm1
0
f(y1,··· , ym,Υ−1
1(α),··· ,Υ−1
n(α))
dαdΨ1(y1)dΨ2(y2)···dΨm(ym).
3 Problem description
In this article, we consider an uncertain random
block renewal model. The so-called block renewal is
a component is always replaced when damaged or re-
placed with a fixed cycle of T. Let’s first assume the
lifetime of a batch of components ξ1, ξ2,··· is a list of
positive independent and identically distributed ran-
dom variables uncertainty variables, and their com-
mon uncertainty distribution is Φ. Its components are
always replaced at a fixed time kT (k= 1,2, ..., ), and
are not replaced when there is a fault, thus maintain-
ing the fault state until the detection of kT at a fixed
time, where Tis a given time period.
Making c1represents the unit time cost incurred
due to component failure during the time period from
the occurrence of the malfunction to the replacement
of the component. c2represents the cost of planning
replacement components at a fixed time.
However, in reality, due to factors such as mar-
ket conditions and management conditions, c1is not
necessarily a fixed constant, but a random variable in-
fluenced by random factors. Therefore, we assume
c1as a random variable to establish an uncertain ran-
dom model. Let c1be a positive random variable with
probability distribution Ψand denote the downtime
cost per unit time elapsed between a failure and its re-
placement, and c2be a constant and denotes the cost
of planned replacement.
4 Uncertain Random Block
Replacement Policy with No
Replacement at Failure
Combining the above problem description, we
consider a multi part system with n independent work-
ing parts, and the lifetime of the components fol-
lows the same uncertainty distribution Φ. For each
part, we assume that it is damaged before the planned
replacement time, it is not replaced and all compo-
nents are replaced at a fixed planned replacement time
kT (k= 1,2, ..., ). Assuming ξ1, ξ2,··· , ξnis the
lifetime of the components. So the cost per unit time
generated during each replacement cycle Tis
C(T, n, c1, ξ1, ξ2,··· , ξn)
=1
Tc1
n
i=1
(T−T∧ξi) + c2.
Given a positive real number T. It is easy to test
and verify that the uncertain variables T∧ξ1, T ∧
ξ2,··· , T ∧ξnhave a common uncertainty distribu-
tion
Υ1(x) = 0,if x < 0
Φ(x),if 0≤x < T
1,if x≥T.
Since
M{T−T∧ξi≤x}
=M{T∧ξi≥T−x}= 1 −Υ1(T−x),
then the uncertain variables T−T∧ξ1, T −T∧
ξ2,··· , T −T∧ξnhave a common uncertainty dis-
tribution Υ2(x), and
Υ2(x) = 0,if x≤0
1−Φ(T−x),if 0< x ≤T
1,if x > T.
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Thus,
E[T−T∧ξi]
=+∞
0
(1 −Υ2(x))dx −0
−∞
Υ2(x)dx
=T
0
Φ(T−x)dx ++∞
T
0dx −0
−∞
0dx
=T
0
Φ(T−x)dx
=T
0
Φ(x)dx.
Recall that the expected cost rate in one cycle
C(T, n, c1, ξ1, ξ2,··· , ξn)is a function of uncertain
variables ξ1, ξ2,··· , ξnand a random variable c1,
thus, C(T, n, c1, ξ1, ξ2,··· , ξn)is an uncertain ran-
dom variable.
In order to find the optimal replacement time T∗,
we give the following theorems.
Theorem 4 Let ξ1, ξ2,··· , ξnbe a sequence of iid
positive uncertain variables with a common uncer-
tainty distribution Φ, and let c1be a positive random
variable with a probability distribution Ψ. Given that
C(T, c1, ξ1, ξ2,··· , ξn)
=1
Tc1
n
i=1
(T−T∧ξi) + c2,
then
E[C(T, n, c1, ξ1, ξ2,··· , ξn)]
=1
TnE[c1]T
0
Φ(x)dx +c2.
Proof: The inverse uncertainty distribution of the
uncertain variable T−T∧ξiis Υ−1
2(α). From the
Theorem 3,
E[C(T, n, c1, ξ1, ξ2,··· , ξn)]
=+∞
01
0
1
Ty
n
i=1
Υ−1
2(α) + c2dαdΨ(y)
=+∞
0
1
T1
0nyΥ−1
2(α) + c2dαdΨ(y)
=+∞
0
1
Tny 1
0
Υ−1
2(α)dα +1
0
c2dαdΨ(y)
=+∞
0
1
T[nyE[T−T∧ξi] + c2]dΨ(y)
=+∞
0
1
Tny T
0
Φ(x)dx +c2dΨ(y)
=+∞
0ny
TT
0
Φ(x)dx +c2
TdΨ(y)
=1
T+∞
0
nydΨ(y)T
0
Φ(x)dx ++∞
0
c2dΨ(y)
=1
TnE[c1]T
0
Φ(x)dx +c2.
The theorem is verified.
Differentiating E[C(T, n, c1, ξ1, ξ2,··· , ξn)] with
respect to Tand setting it equal to zero, we can get
nTΦ(T)−T
0
Φ(x)dx=c2
E[c1].(1)
Thus, if there exists an optimum time T∗that
uniquely determined by the equation (1), the result-
ing expected cost per unit of time is
E[C(T, n, c1, ξ1, ξ2,··· , ξn)] = nE[c1]Φ(T∗).
Next, we discuss the existence of T∗.
Theorem 5 Let ξibe a sequence of positive uncertain
variables with a common uncertainty distribution Φ,
and let c1be a positive random variable with a prob-
ability distribution Ψ. Given that
C(T, c1, ξ1, ξ2,··· , ξn)
=1
Tc1
n
i=1
(T−T∧ξi) + c2,
If
lim
T→+∞nTΦ(T)−T
0
Φ(x)dx>c2
E[c1],(2)
then there exists a finite and unique T∗(0 < T ∗<
+∞)that satisfies
nT∗Φ(T∗)−T∗
0
Φ(x)dx=c2
E[c1].(3)
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Proof: Let
Q(T) = nTΦ(T)−T
0
Φ(x)dx.
It is lightly to obtain that lim
T→0Q(T) = Q(0) = 0.
And Q(T)is increasing because for any ∆T > 0,
Q(T+ ∆T)−Q(T)
=n[xΦ(x)]T+∆T
T−T+∆T
T
Φ(x)dx>0.
If Q(+∞)>c2
E[c1], then, from the monotonicity and
the continuity of Q(T), there exists a finite and unique
T∗(0 < T ∗<+∞)that satisfies equation (1) and
minimizes E[C(T, n, c1, ξ1, ξ2,··· , ξn)].
The theorem is verified.
5 Numerical example
Ceiling lamp in the terminal is an important com-
ponent part of the lighting system of an airport. For
the sake of saving source of energy in a long time, a
new type of LED lamps are adopted to displace old
fluorescent bulbs in the terminal, and take the block
replacement policy with no replacement at failure to
this lighting system. In this section, we will find the
optimal replacement time of the new lighting system
for the airport.
The data of the lifetimes of LED lamps are un-
available because this kind of LED light is fresh and
never used in any place, then the lifetimes are as-
sumed to be independent uncertain variables with a
linear uncertainty distribution Φ(x):
Φ(x) = 0,if x≤a
(x−a)/(b−a),if a≤x≤b
1,if x≥b.
In this case, we invite some industry experts to
evaluate their possible lifetime and reliability. Based
on the subjective reliability of experts and the least
squares method in uncertainty theory, [7]. The pa-
rameter values in the uncertain distribution can be ob-
tained separately ˆa= 20,000,ˆ
b= 50,000.
Assuming a set of ceiling lights consists of n=
1800LED lights that operate independently of each
other, the cost-per-install is c2= $34,600. If the re-
placement cycle is T, when any one lamp fails in the
interval (kT, (k+1)T), k = 0,1,···, the cost per unit
time c1caused by various kinds of salvage is supposed
to be a random variable with normal distribution
P(x) = 1
σ√2πx
−∞
exp −(t−µ)2
2σ2dt,
and the estimate of expected value ˆµ= 11.
After a verification, the condition that
lim
T→+∞nTΦ(T)−T
0
Φ(x)dx>c2
µ(4)
is satisfied.
Therefore, according to the equation (3), the opti-
mal block replacement time is
T∗=a2+2c2(b−a)
nµ = 20002.62.
That is to say, this set of light bulbs needs to be
replaced every 20002.62 hours.
0 1 2 3 4 5 6 7 8 9 10
x 106
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
b−a
T*
(i)
5 6 7 8 9 10 11 12 13 14 15
1502
1503
1504
1505
1506
1507
1508
µ
T*
(ii)
0 0.5 1 1.5 2 2.5
x 104
1500
2000
2500
3000
3500
4000
4500
T*
c2/n
(iii)
Fig. 1: Optimal replacement time T∗for increasing
values of (i) b−a, (ii) µ, (iii) c2/n.
Fig.1(i) can find that the optimal replacement time
T∗shows an increasing trend with the increase of b−
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a(for fixed a). It can be explained that the increase
in bdollars means an extension of the unit’s lifetime,
leading to longer replacement cycles.
From Fig.1(ii) we can see that when µincreases,
the optimal replacement time T∗gradually decreases
and shows a decreasing trend. In practice, if the
downtime cost per unit time is too high, to reduce the
downtime cost, we should lower the planned replace-
ment time T∗. We also observe that T∗follows value
of c1increases and decreases more slowly.
Fig.1(iii) shows that the optimal replacement time
T∗decreases as c2/nincreases. That is to say, if the
planned replacement cost is too high, the planned re-
placement time should be increased to cut down the
cost rate within a replacement cycle.
6 Conclusion
The paper considered a block replacement policy
with no replacement at failure in uncertain random en-
vironment, due to the fact that the components in the
system are brand new, there is no historical life data
available, in which the lifetimes of components were
assumed as uncertain variables and the downtime cost
per unit time elapsed between a failure and its replace-
ment was considered to be a random variable. In order
to deal with the coexistence of uncertainty and ran-
domness, we developed an uncertain random block
replacement model with no replacement at failure for
a system to minimize the expected cost per unit time
in one replacement cycle in uncertain random frame-
work. We derived an equation which determined the
optimal replacement time and also gave the condition
to ensure the existence and uniqueness of the optimal
block replacement time. The application of our model
in practice was illustrated by a numerical example.
The results of sensitivity analysis indicated that the
optimal replacement time was increasing with respect
to the difference between the maximum lifetime and
the minimum lifetime of the unit, and the planned re-
placement cost, and decreased in the downtime cost
per unit time.
In the future, we will concentrate on the following
research:
(1) In the future, if there is sufficient data and c1is
a deterministic variable, we will consider using ran-
dom theory methods for research.
(2) Future research can be continued by consider-
ing minimal repairs together with block replacement
in uncertain random environment. The degradation
process of units could be taken into account to make
the model more realistic.
7 Application
This paper presents a model with uncertain ran-
dom variables, its research process and results can
provide maintenance theoretical support for airlines
that lack complete operational data.
Acknowledgements
The authors would like to thank the editors
and reviewers for their valuable comments on the
manuscript.
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Contribution of individual authors to
the creation of a scientific article
Zheng Wu proposed the idea of the model and
checked the correctness of the manuscript.
Xinwang Li established the mathematical model and
wrote the article.
Min Feng and Hejiao Shen gave a numerical example
and wrote the article.
Chunxiao Zhang checked the logic of the article and
the coherence of the language.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is supported by the Fundamental Research
Funds for the Central Universities under Grant
No.3122014D034.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.69
Zheng Wu, Xinwang Li, Min Feng, Hejiao Shen, Chunxiao Zhang
E-ISSN: 2224-2880
633
Volume 22, 2023
