Maintenance Policy with Lifetime Reduction Based on Uncertain

Renewal Process

CHUNXIAO ZHANG, XIAONA LIU, XINWANG LI, YIZHOU BAI

College of Science

Civil Aviation University of China

Tianjin 300300

CHINA

Abstract: This paper focus on a (N, T )preventive maintenance (PM) policy with lifetime reduction discount

rate. In our consideration, the lifetime of component is assumed to be an uncertain variable due to the absence of

historical operational data, an uncertain (N, T )PM model with the lifetime reduction in a certain proportion is

proposed based on uncertain renewal process. Accordingly, the uncertain model, which minimizes the expected

maintenance cost rate, is formulated to find the optimal replacement cycle length T∗and the number N∗of pre-

ventive maintenance. Finally, a numerical example with sensitivity analysis of parameters is provided to illustrate

the proposed model, the results imply that the parameters of cost and lifetime can significantly affect the optimal

solutions N∗and T∗, which can provide a useful reference and guidance for aircraft maintenance decision.

Key-Words: -(N, T )preventive maintenance policy, Minimal repair, Lifetime reduction, Uncertain renewal

process.

Received: July 17, 2022. Revised: October 5, 2022. Accepted: November 6, 2022. Available online: December 19, 2022.

1 Introduction

Preventive maintenance (PM) mainly refers to the

maintenance method of carrying out a series of main-

tenance on the premise that the mechanical equip-

ment has no failure or damage, and all activities are

carried out to prevent functional failure and keep it

in the specified state through systematic inspection,

maintenance and replacement of products. There-

fore, it is usually used in engineering areas to avoid

the consequences of failure endangering safety, af-

fecting the completion of tasks or causing major eco-

nomic losses. Barlow and Hunter, [19], first pro-

posed preventive maintenance policy to reduce main-

tenance costs, together with risks and losses caused

by accidents. Kulshrestha considered the reliability

of preventive maintenance, and studied the system

with preventive maintenance under different distribu-

tions, [9]. Beichelt, [10], gave the optimum preven-

tive maintenance of systems, which were age replace-

ment policy and minimal repair policy, when the sys-

tem may have two kinds of faults.

Over time, maintenance issues are attracting more

and more attention. Boland and Proschan, [18], pro-

posed to replace or repair regularly at a fixed multiple

of the predetermined time T, and minimal repair if

the item failed. And in [21], Nakagawa presented the

minimal repair strategy in the age replacement pol-

icy. If the components failed before T, minimal re-

pair was carried out. When the components failed in

the fixed cycle T, we carried out the preventive re-

placement. A repairable item system, performance

of which was measured by a service level, was intro-

duced by De Haas and Verrijdt, [12], for the main-

tenance support of a flet of aircraft. In [22], Makis

and Cheng considered a repair/replacement problem

for a single unit system with random repair cost.

When the component failed, we decided whether to

replace or repair it by observing the repair cost. Park

et al., [17], developed a renewable minimal repair-

replacement warranty policy and proposed an opti-

mal maintenance model after the warranty expires.

In [20], Gopalan proposed an inverse optimization

model, and performed all required aircraft mainte-

nance activities with a stipulated periodicity. Safaei et

al., [11], proposed a policy to determine whether a re-

pairable system should be repaired or replaced with a

new system when it failed in the optimization of main-

tenance policy.

As mentioned in the previous literatures, the life-

times of the components were usually assumed as a

stochastic variable. As we know, only when there

are enough samples can the knowledge of probabil-

ity theory be applied. In this paper, the components

we focus on are made of new materials, and have

few or even no historical operating data of their life-

time. In this case, we usually invite some domain ex-

perts to provide belief degree of lifetime to estimate

its distribution. However, human often tends to over-

estimate unlikely events ( Tversky and Kahneman in

[13]), which make the variance of belief degree much

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greater than that of the frequency, and the probability

theory is no longer suitable. To deal with this kind of

problem, uncertainty theory was founded by Liu, [1],

then refined by Liu, [3], which has become a branch

of axiomatic mathematics.

At present, the uncertainty theory based on Profes-

sor Liu has been further developed and applied in var-

ious fields of practical production practice. Liu in [2]

provided an independent and comprehensive uncer-

tain programming theory to solve modeling optimiza-

tion problems in uncertain environments. In [24], Liu

and Ha proved the expected values of monotone func-

tions with uncertain variables and gave some useful

expressions for the expected values of functions with

uncertain variables. In the same year, Liu, [3], de-

fined an uncertain renewal process, in which events

occurred continuously and independently of one an-

other in uncertain times. Next, Yao and Li, [14], pro-

posed an uncertain alternating renewal process, which

its alternating arrival time was an uncertain variable.

Then, several researchers applied the uncertainty

renewal process in maintenance areas. Yao and Dan,

[15], assumed that the lifetime of component was an

uncertain variable based on the age replacement pol-

icy involving random age, and found out the optimal

time to replace the component. Zhang and Guo, [5],

discussed the optimal age replacement policy for the

lifetime of components in different uncertainty distri-

butions. Next year, Zhang and Guo in [6] proposed an

uncertain block replacement model without replace-

ment in case of failure, and the existence conditions of

the optimal replacement time were given. Yao, [16],

gave an uncertain block replacement policy, assum-

ing that the lifetime of the components was an uncer-

tain variable. Recently, based on the uncertainty the-

ory, an uncertain (N, T )block replacement policy for

aircraft structures with corrosion damage was studied

by Zhang and Li, [7]. Zhang and Li, [8], proposed

an extended uncertain stochastic renewal reward the-

orem, and established an uncertain random program-

ming model with the objective of minimizing the ex-

pected maintenance cost rate.

In this paper, we consider a (N, T )preventive

maintenance policy with lifetime reduction discount

rate. The policy means that a component is carried out

preventive maintenance at the fixed time KT (K=

1,2, . . . , N)symmetrically, and after each imperfect

preventive maintenance, the lifetime of component

will be reduced by β(0 ≤β≤1), where βis

a specific number. We only carry out minimal re-

pairs between preventive maintenance KT and (K+

1)T(K= 1,2, . . . , N), then this component is re-

placed by a new one at the time NT (N= 1,2, . . .).

Assuming that the lifetime of new component is an

uncertain variable due to no historical operation data,

we will develop an uncertain (N, T )preventive main-

tenance model that minimizes the expected mainte-

nance cost based on the uncertain renewal process,

where the renewal cycle length Tand the number N

of preventive maintenance are decision variables.

The rest of this paper is organized as follows. In

Section 2, we make the problem description. In Sec-

tion 3 we build a uncertain (N, T )preventive mainte-

nance model in uncertain circumstance, and analyze

its optimal solution. A numerical example with pa-

rameter sensitivity analysis is provided, which illus-

trates the usefulness of the proposed model in Section

4. Finally, the Section 5 presents some conclusions.

2 Problem description

In this section, we consider a uncertain (N, T )

preventive maintenance policy with lifetime reduc-

tion discount rate, where the lifetime of component

is regarded as an uncertain variable. The component

needs to carry out preventive maintenance at the fixed

time KT (K= 1,2, . . . , N ). After each imperfect

preventive maintenance, the lifetime of component

will deduce by xunits. Then, preventive replacement

is made at the Nth preventive maintenance. Here, we

assume that the failure rate is reduced to 0 after mini-

mal repair, each PM is an imperfect preventive main-

tenance and the time of minimal repair and preventive

maintenance are negligible. The preventive mainte-

nance policy we proposed can be illustrated using the

notations in Fig.1.

Fig.1: Preventive maintenance policy based on

lifetime discount rate (x= (1 −β)ξ)

In order to better establish the model, some nota-

tions are made as follows:

T: the replacement cycle length of preventive

maintenance;

N: the number of preventive maintenance;

ξk: the lifetime of components in the kth(k=

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1,2, . . . , N)preventive maintenance cycle, is an un-

certain variable;

β: the percentage of component lifetime reduction

after each preventive maintenance(0 ≤β≤1), then

the lifetime of the component after the kth preventive

maintenance is ξk=βk−1ξ1(K= 1,2, . . . , N);

Φ(t): the uncertainty distribution function of

structural component lifetime ξ1in the first preven-

tive maintenance cycle;

M(k): the number of minimal repair from the

(k−1)th to the kth preventive maintenance, is an

uncertain renewal process;

c1: cost of minimal repair of components before

time T;

c2: cost of preventive maintenance of components

at time T;

c3: replacement cost of components at time N T ,

c3> c2.

3 Mathematical formulation

According to the above notations, the total cost of

the whole preventive renewal cycle can be expressed

T C(N, T, ξ) = c1



k=1

M(k) + c2(N−1) + c3(1)

Then, based on the above analysis, the objective

function is

C(N, T, ξ) =



k=1

M(k) + c2(N−1) + c3

NT (2)

where c1



k=1

M(k)is total minimal repair cost,

c2(N−1) is total cost of preventive maintenance, c3

is the cost of planned replacement at time NT .

Recall ξis an uncertain variable, the cost rate func-

tion C(N, T, ξ)is an uncertain variable as well, so it

cannot be directly minimized with respect to Nand

T, and we may minimize its expected value:

C(N;T) = min

N,T E[C(N, T, ξ)](3)

That is, the optimal N, T should be solved by the

following model:

min

N,T

c1E(



k=1

M(k)) + c2(N−1)+c3

(4)

In order to solve the model (4), we present the fol-

lowing lemmas:

Lemma 3.1 [23] Let fand gbe comonotonic func-

tions. Then for any uncertain variables ξ, we have

E[f(ξ) + g(ξ)] = E[f(ξ)] + E[g(ξ)] (5)

And then, the Lemma 3.1 can be extended to a limited

number.

Lemma 3.2 [1] Let ξbe an uncertain variable with

uncertainty distribution Φ(x). Then

E(ξ) = +∞

(1 −Φ (x))dx −0

−∞

Φ (x)dx (6)

Lemma 3.3 [3] Let M(k)be a renewal process with

iid positive uncertain interarrival times ξ1, ξ2, ξ3, . . ..

If Φkis the common regular uncertainty distribution

of those interarrival times, then M(k)has an uncer-

tainty distribution

γk(x) = 1 −Φk(T

⌊x⌋+ 1),∀x≥0(7)

where ⌊x⌋represents the maximal integer less than or

equal to x.

Since M(k)(k= 1,2, . . . , N)is respectively a

subtractive function of ξk(k= 1,2, . . . , N), accord-

ing to Lemma 3.1, we have:



k=1

M(k)] =



k=1

E[M(k)] (8)

From Lemma 3.2 and Lemma 3.3:

E[M(k)] = ∞

Φk(T

⌊x⌋+ 1)dx (9)

For ξk=βk−1ξ1, it has

Φk(x) = M(ξk≤x)

=Mβk−1ξ1≤x

=Mξ1≤β1−kx

= Φ x

βk−1, k > 1

(10)

then,

E[M(k)] = ∞

Φk(T

⌊x⌋+ 1)dx

=∞

Φ( T

⌊xβ1−k⌋+ 1)dx

(11)

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Substitute E[M(k)] into (4), the model is trans-

formed into:

C(N;T) = min

N,T E[C(N, T, ξ)]

=min

N,T

c1E(



k=1

M(k)) + c2(N−1)+c3

=min

N,T



k=1 ∞

0Φ( T

⌊xβ1−k⌋+1 )dx +c2(N−1)+c3

(12)

Next, we discuss the optimal solution of the

model:

For specified T > 0and β, we seek an optimum

replacement number N∗(1 ≤N∗≤ ∞)that mini-

mizes C(N, T ). From the inequality C(N+ 1; T)≥

C(N;T), that is

N+1



k=1 ∞

0Φ( T

⌊xβ1−k⌋+1 )dx +c2N+c3

(N+ 1) T

≥



k=1 ∞

0Φ( T

⌊xβ1−k⌋+1 )dx +c2(N−1)+c3

NT (13)

We have,

L(N;T)≥(c3−c2)

(N= 1,2,···)(14)

where,

L(N;T) = N

∞



ΦT

⌊xβ−N⌋+ 1dx

−



k=1

∞



ΦT

⌊xβ1−k⌋+ 1dx



k=1

∞



ΦT

⌊xβ−N⌋+ 1−ΦT

⌊xβ1−k⌋+ 1dx



k=1

∞



Φ T

⌊xβ−N⌋+ 1−T

⌊xβ1−k⌋+ 1dx

(15)

In addition, we have

L(N+ 1; T)−L(N;T)

N+1



k=1

∞



Φ T

⌊xβ−N−1⌋+ 1−T

⌊xβ1−k⌋+ 1dx

−



k=1

∞



Φ T

⌊xβ−N⌋+ 1−T

⌊xβ1−k⌋+ 1dx

= (N+ 1)

∞



Φ T

⌊xβ−N−1⌋+ 1−T

⌊xβ−N⌋+ 1dx

(16)

It can be seen that the above formula (16) is always

greater than 0. So L(N;T), when Tand βare speci-

fied, is a constant increasing function with respect to

Therefore, we have the following optimum policy:

(i) If L(∞;T)≡limN→∞L(N;T)>

(c3−c2)/c1, then there exists a finite and unique

minimum N∗that satisfies (14).

(ii) If L(∞;T)≤(c3−c2)/c1, then N∗=∞,

and the expected cost rate is

C(∞;T) = lim

N→∞ C(N;T)

=lim

N→∞



k=1 ∞

0Φ( T

⌊xβ1−k⌋+1 )dx +c2(N−1)+c3

=c1lim

N→∞



k=1 ∞

0Φ( T

⌊xβ1−k⌋+1 )dx

NT +c2

(17)

4 Numerical example

In this section, we apply our policy to the mainte-

nance of a new kind of aircraft parts, such as cargo

bulkhead, which is made by a new developed alu-

minum alloy. The data of the lifetime are unavailable,

and as a result, the lifetime is regarded as a uncertain

variable with regular uncertainty distribution. We in-

vite some experts in the field of cargo bulkhead to pro-

vide possible lifetime and belief degrees. According

to the experts’ belief, we find that the lifetime of cargo

bulkhead follows a normal uncertain distribution, [1].

And then the uncertainty distribution function of the

lifetime ξ1is:

Φ (x) = 1 + exp π(e−x)

√3σ−1

, x ∈ ℜ (18)

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Then,

C(N;T)

=min

N,T



k=1 ∞

0Φ( T

⌊xβ1−k⌋+1 )dx +c2(N−1)+c3

=min

N,T



k=1 ∞

0

1 + exp 



π(e−T

⌊xβ1−k⌋+1 )

√3σ





−1

+c2(N−1)+c3

NT (19)

According to the principle of least squares given

by Liu in [4], we obtain the estimated values of un-

known parameters of the uncertainty distribution: ˆe=

10,ˆσ=√3. Let the minimal repair cost of compo-

nent c1= $1.5, cost of preventive maintenance c2=

$5, and the cost of planned replacement c3= $25.

The optimal solutions of the number N∗and replace-

ment cycle length T∗of preventive maintenance are

obtained for minimizing the expected cost function

C(N;T), and shown in Fig.2.

Fig.2: Optimal replacement cycle length T∗and the

number N∗of preventive maintenance

It can be found from the Fig.2 that the objec-

tive function is a convex function, that is, there is

an optimal solution. We can obtain the minimal cost

C∗(N;T)when N∗= 6 and T∗= 3.2. So, the air-

craft cargo bulkhead shoud be subject to preventive

maintenance every 3.2 years, and can be planned to

be replaced at the 6th preventive maintenance. Then,

the minimal preventive maintenance cost is calculated

to be $2.604.

The sensitivity analysis of parameters in our model

are given in Fig.3.

(i)

(ii)

(iii)

(iv)

Fig.3: Optimal replacement cycle length T∗for

increasing values of (i) c1, (ii) c2, (iii) c3, (iv) e

Analyzing from Fig.3 (i) to (iii), we can see that

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the optimal value is significantly sensitive with re-

spect to cost factor. Fig.3 (i) shows that the optimal

replacement cycle length T∗is decreasing with the

growth of the cost of minimal repair c1, and decreases

more slowly with an increasing value of c1, which in-

dicates that the replacement cycle length should be

shortened when the cost of minimal repair is too high.

Fig.3 (ii), (iii) indicate that the optimal replacement

cycle length T∗is increasing with the cost of preven-

tive maintenance c2increases, and when the replace-

ment cost c3is large enough, T∗will increase, too. It

can be interpreted that the replacement cycle length

should be extended properly to reduce the cost rate

if the cost of preventive maintenance c2and the re-

placement cost c3are too much in a replacement cy-

cle. From Fig.3 (iv), the optimal replacement cycle

length T∗increases when the ebecomes large.

Meanwhile, in Fig.3 (i) to (iii), the optimal re-

placement cycle length T∗is inversely proportional

to the number of preventive maintenance N∗, that is,

when the number N∗is increasing, the optimal re-

placement cycle length T∗is on a downtrend. How-

ever, in Fig.3 (iv), the optimal replacement cycle

length T∗is proportional to the number of preventive

maintenance N∗.

5 Conclusion

This paper studied the preventive maintenance

policy with lifetime reduction discount rate based on

the uncertain renewal process. A kind of component

which was made by a new developed composite ma-

terial, so there was no historical data to obtain the

probability distribution of the lifetime, the stochastic

method was not applicable for our research yet. Thus,

the lifetime of component was regarded as an uncer-

tain variable, and considering that the lifetime of com-

ponent would reduce by a fixed unit after each imper-

fect preventive maintenance. Next, we took the pre-

ventive maintenance cycle Tand number Nas the de-

cision variables, the minimal expected cost rate as the

objective function. Then, a preventive maintenance

model with the lifetime reduction discount rate was

established. Finally, a numerical example was given

and the sensitivity analysis of the parameters were

made. The results showed that the optimal preven-

tive replacement cycle length T∗would decrease with

the increase of minimal repair cost of component, and

increase with the increase of preventive maintenance

cost and replacement cost.

In the future works, we may focus on the following

issues:

(i) In this work, the time of minimal repair and

preventive maintenance are negligible. Next, we can

consider the situation that its time can not be ignored

in the future.

(ii) Also, in this work, we assumed that the life-

time of component follows an uncertain distribution

due to no available lifetime data. Furthermore, when

the operational data is fully obtained, some random

lifetime distributions for the original system lifetime

can be considered, such as weibull or normal distri-

bution.

Acknowledgements

The authors would like to thank the editors

and reviewers for their valuable comments on the

manuscript.

References:

[1] B. Liu, Uncertainty Theory, 2th ed., Springer-

Verlag, Berlin, 2007.

[2] B. Liu, Theory and Practice of Uncertain Pro-

gramming, 2nd ed., Springer-Verlag, Berlin,

2009.

[3] B. Liu, Uncertainty Theory: A Branch of

Mathematics for Modeling Human Uncertainty,

Springer-Verlag, Berlin, 2010.

[4] B. Liu, Uncertainty Theory, 4th ed., Uncertainty

Theory Laboratory, 2013.

[5] C. X. Zhang, C. R. Guo, Some new results on un-

certain age replacement policy, Industrial Engi-

neering and Management Systems, Vol.12, No.1,

2013, pp.41-45.

[6] C. X. Zhang, C. R. Guo, Uncertain block re-

placement policy with no replacement at failure,

Journal of Intelligent and Fuzzy Systems: Appli-

cations in Engineering and Technology, Vol.27,

No.4, 2014, pp.1991-1997.

[7] C. X. Zhang, Q. Y. Li, X. L. Shi, Uncertain

(N, T )block replacement policy of aircraft struc-

ture subjected to corrosion damage, Soft Comput-

ing, Vol.20, No.11, 2016, pp.4619-4627.

[8] C. X. Zhang, X. W. Li, X. N. Liu, Q. Li, Y.

Z. Bai, Repair limit policy of aircraft compo-

nent based on extended uncertain random re-

newal reward process, Engineering Computa-

tions, Vol.39, No.6, 2022, pp.2033-2052.

[9] D. K. Kulshrestha, Reliability with preven-

tive maintenance, Metrika, Vol.19, No.1, 1972,

pp.216-226.

[10] F. Beichelt, A general preventive maintenance

policy, Optimization, Vol.7, No.6, 1976, pp.927-

932.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.4

Chunxiao Zhang, Xiaona Liu, Xinwang Li, Yizhou Bai

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[11] F. Safaei, J. Ahmadi, N. Balakrishnan, A repair

and replacement policy for repairable systems

based on probability and mean of profits, Reli-

ability Engineering and System Safety, Vol.183,

2019, pp.143-152.

[12] H. De, J. Verrijdt, Target setting for the depart-

ments in an aircraft repairable item system, Eu-

ropean Journal of Operational Research, Vol.99,

No.3, 1997, pp.596-602.

[13] K. A. Tversky, Prospect Theory: An Analysis

of Decision under Risk, Econometrica, Vol.47,

No.2, 1979, pp.263-291.

[14] K. Yao, X. Li, Uncertain alternating renewal

process and its application, IEEE Transactions

on Fuzzy Systems, Vol.20, No.5, 2012, pp.1154-

1160.

[15] K. Yao, A. R. Dan, Age replacement policy in

uncertain environment, Iranian Journal of Fuzzy

Systems, Vol.10, No.2, 2013, pp.29-39.

[16] K. Yao, Block replacement policy in uncertain

environment, Reliability Engineering and System

Safety, Vol.148, 2016, pp.119-124.

[17] M. Park, K. M. Jung, H. P. Dong, Optimal post-

warranty maintenance policy with repair time

threshold for minimal repair, Reliability Engi-

neering &System Safety, Vol.111, 2013, pp.147-

153.

[18] P. J. Boland, F. Proschan, Periodic replace-

ment with increasing minimal repair costs at fail-

ure, Operations Research, Vol.30, No.6, 1982,

pp.1183-1189.

[19] R. Barlow, L. Hunter, Optimum preven-

tive maintenance policies, Operations Research,

Vol.8, No.1, 1960, pp.90-100.

[20] R. Gopalan, The Aircraft Maintenance Base

Location Problem, European Journal of Oper-

ational Research, Vol.236, No.2, 2014, pp.634-

642.

[21] T. Nakagawa, Sequential Imperfect Preventive

Maintenance Policies, RTransaction on Reliabil-

ity, Vol.37, No.3, 1988, pp.295-298.

[22] V. Makis, X. J. Cheng, Optimal Preventive

Replacement Under Minimal Repair and Ran-

dom Repair Cost, Mathematics of Operations Re-

search, Vol.25, No.1, 2000, pp.141-156.

[23] X. H. Yang, On comonotonic functions of uncer-

tain variables, Fuzzy Optim Decis Mak, Vol.12,

No.1, 2013, pp.89-98.

[24] Y. H. Liu, M. H. Ha, Expected value of function

of uncertain variables, Journal of Uncertain Sys-

tems, Vol.4, No.3, 2010, pp.181-186.

Contribution of individual authors to

the creation of a scientific article

Chunxiao Zhang proposed the idea of the model and

checked the correctness of the manuscript.

Xiaona Liu established the mathematical model and

gave the existence conditions of the solution.

Xinwang Li gave an numerical example and wrote

the article.

Yizhou Bai 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 Open Fund of Civil

Aviation University of China for Provincial and

Ministerial Scientific Research Institutions under

Grant No.TKLAM202201.

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DOI: 10.37394/23206.2023.22.4

Chunxiao Zhang, Xiaona Liu, Xinwang Li, Yizhou Bai

E-ISSN: 2224-2880

Volume 22, 2023

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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(Attribution 4.0 International, CC BY 4.0)

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Creative Commons Attribution License 4.0

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