AN AGE-BASED MAINTENANCE STRATEGY FOR A DEGRADATION-THRESHOLD-SHOCK-MODEL FOR A SYSTEM SUBJECTED TO MULTIPLE DEFECTS

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August 20, 2013 16:27 WSPC/S0217-5959 APJOR 1350016.tex

Asia-Pacific Journal of Operational ResearchVol. 30, No. 5 (2013) 1350016 (14 pages)c© World Scientific Publishing Co. & Operational Research Society of SingaporeDOI: 10.1142/S0217595913500164

AN AGE-BASED MAINTENANCE STRATEGYFOR A DEGRADATION-THRESHOLD-SHOCK-MODELFOR A SYSTEM SUBJECTED TO MULTIPLE DEFECTS

I. T. CASTRO∗

Department of MathematicsUniversity of Extremadura

10071 Caceres, [email protected]

Received 4 October 2010Accepted 22 May 2013

Published 22 August 2013

A system subject to internal defects and external shocks is analyzed in this paper.Internal defects initiate following a nonhomogeneous Poisson process (NHPP) and theygrow according to deterioration processes modeled as gamma processes. A correctivereplacement is performed when the deterioration of a defect exceeds a failure threshold.The system is subject to external shocks. After an external shock, the system is replacedwith probability 1 − p and is minimally repaired with probability p. The system ispreventively replaced at the age of T . Costs are associated with the maintenance actions.The value of T that minimizes the expected cost rate is obtained analytically. Numericalexamples are showed to illustrate the theoretical results.

Keywords: Age based maintenance; degradation-threshold-shock models; gamma pro-cess; pitting corrosion; nonhomogeneous Poisson process; minimal repair.

1. Introduction

With the development of the industrial systems, maintenance plays an importantrole directly related to the competitiveness of the companies. In practice, manysystems degrade physically over time and the failure mechanisms can be traced toan underlying degradation process (internal wear, material fatigue, crack-growth).Such degradation process can be modeled using a stochastic process. Accordingto Lehmann (2009), the stochastic-processes-based approach shows great flexibil-ity to describe the failure-generating mechanisms and can give alternative time-to-failure-distributions defined by the degradation model. Frequently, under thisapproach, the system is regarded as failed and it is switched off when its degrada-tion first reaches a critical threshold level.

∗Corresponding author

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Besides degradation failures, systems may also be subject to external shockswhich may lead to failure. Lemoine and Wenocur (1985) may have been the firsts toconsider these two competing causes of failures (degradation reaching a threshold oroccurrence of a traumatic shock) and these models are called degradation-threshold-shock- models (DTS-models). So, the failure time of a system is the minimum of themoment when degradation first reaches a critical threshold and the moment when acensoring traumatic event occurs. Cox (1999) and Singpurwalla (1995) give detailedreviews on stochastic-process-based reliability models including DTS-models. Somepractical examples are found in the literature. Meeker and Escobar (1998) describeda practical case of DTS model as the life of some laser devices. The degradation of alaser device causes an increase in its operating current. When operating current getstoo high, the device is considered to have failed. Besides degradation, inadvertentshocks may also result in a system failure. Ye et al. (2011) showed as practicalcase of DTS model the operating life of an ammeter. The ammeter degrades overtime and this degradation manifests an increasing resistance. When the resistanceexceeds a threshold, a degradation failure occurs. During the operating life, shockssuch as lightning would destroy the ammeter.

Frequently, in the deteriorating systems literature, the degradation of the systemis an unique measure modeled as a stochastic process. However, in some practicalsituations, the system is subject to multiple defects that weak its resistance. Anexample of this situation is the pitting corrosion process. Pitting corrosion is a typeof corrosion that leads to the creation of small corrosion pits on the surface of somemetals. Furthermore, each pit follows a growth process and can penetrate inwardsextremely rapidly causing the failure of an engineering system. The pitting corrosionmodel can be considered as a combination of two processes: pit initiation and pitgrowth process and is a predominant form of degradation of steam generators ofnuclear power plants (Datla, 2007). It is widely recognized that pitting corrosionhas a stochastic nature (Williams et al., 1985). Recently, Kuniewski et al. (2009)showed a probabilistic model representing the pitting corrosion process and theyused a Bayesian method for inference on the number of pits in the entire systembased on partial inspection. Castro et al. (2011) developed a stochastic model fora pitting corrosion process assuming that the pits are detectable when their depthexceeds a predetermined value.

This paper adds a new dimension to the pitting corrosion models analyzing anage-based maintenance strategy for systems subject to a pitting corrosion processand external shocks. Internal defects (or pits) initiate at random times following anonhomogeneous Poisson process (NHPP). Given that a defect has occurred, theprocess of degradation or growth is activated. Because of its modeling capacityand its mathematical tractability, the stationary gamma process is considered tomodel the degradation of each defect. The stationary gamma process is a stochas-tic process with independent non-negative increments having a gamma distributionwith identical scale parameter. It is a particular case of the general degradationpath model showed by Jiang (2010). The gamma processes has been satisfactorily

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An Age-Based Maintenance Strategy

fitted to data of different gradual degradation phenomena such as erosion, corrosion,concrete creep, crack growth or wear of structural components (Blain et al., 2007;Cinlar et al., 1977; Frangopol et al., 2004). Moreover, the existence of an explicitprobability distribution function of the gamma process permits feasible mathemat-ical developments. Therefore, since the initial proposal by Abdel-Hameed (1975),the gamma process has been analyzed for different maintenance applications by sev-eral authors (see van Noortwijk (2009) for a thorough review on the use of gammaprocess in maintenance modeling).

In this paper we assume that external shocks arrive to the system and they arecatastrophic with probability 1− p and noncatastrophic with probability p. After anoncatastrophic shock, a minimal repair is performed. Minimal repair means thatthe age and the degradation of the system is not disturbed for the occurrence of non-catastrophic shocks. Consequently the failure rate of the noncatastrophic shocks attime t is independent of the number of noncatastrophic shocks occurred up to time t.After a degradation failure or after a catastrophic shock, a corrective maintenanceis performed and the system is replaced by a new one. An age-based maintenancestrategy is developed in this paper. Under this strategy, a preventive maintenance isperformed when the age of the system exceeds a predetermined value T and thesystem is replaced by a new one.

Costs are associated with the different maintenance actions and the objective isto determine an optimal preventive replacement time T . “Optimal” means a valueof T that minimizes an objective cost function. Let {X(t), t ≥ 0} be a regenerativeprocess, we denote by S1, S2, S3, . . . the regeneration epochs of the process and byLi = Si − Si−1, i = 1, 2, . . . the length of the ith renewal cycle. If a cost structureis imposed on the regenerative process {X(t), t ≥ 0} and denoting by C(t) the costof the system at time t and by Ci the total cost in the ith renewal cycle, it is wellknown that (see Tijms, 2003, pp. 41)

C(t)t

→ E[C1]E[L1]

, (1)

with probability one, that is, the long-run average cost per unit time is equal tothe expected cost in a cycle divided by the expected length of the cycle for almostany realization of the process. The criterion long-run average cost is widely usedin reliability literature and we shall use (1) as objective function to analyze theoptimal strategy for this maintenance model.

This paper is structured as follows. Section 2 models the different competingfailure modes of the system and analyzes the associated failure times. The detaileddescription of the maintenance strategy is represented in Sec. 3. Section 4 shows anumerical example of this maintenance strategy and Sec. 5 concludes.

2. Problem Definition and Mathematical Formulation

This paper analyzes a single-unit system whose failures are due to the competingcauses of degradation and shocks. The system is described by a so-called DTS model.

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As Singpurwalla explained (Singpurwalla, 1995), such a failure/deterioration modelcan be seen as a combination and more versatile extension of many classical failuremodels based either only on deterioration or only on parametric lifetime distribu-tions. In this paper, the deterioration is modeled using different time-dependentstochastic processes, and the system is regarded as failed when one of these pro-cesses reaches a critical threshold or when a shock occurs although the degradationprocesses have not reached the threshold.

The assumptions of the model are the following.

(1) The system is subject to different internal defects. These defects initiate atrandom times and they grow depending on the environment and conditions ofthe component. We assume in this paper that internal defects arrive accordingto a NHPP {Nd(t), t ≥ 0} with intensity m(t) and cumulative intensity

M(t) =∫ t

0

m(u)du, t ≥ 0, (2)

where t is the age of the system and m(t) is a nondecreasing function in t.The deterioration of these defects is modeled according to gamma processes. Adegradation failure occurs when the deterioration level of a given defect exceedsthe threshold L.

(2) External shocks arrive according to a NHPP {Ns(t), t ≥ 0} with intensity h(t)and cumulative intensity H(t) given by

H(t) =∫ t

0

h(u)du, t ≥ 0,

where t is the age of the system and h(t) is a nondecreasing function in t.External shocks are noncatastrophic with probability p and catastrophic withprobability 1 − p (0 < p < 1). After a noncatastrophic shock, a minimal repairis performed. After a catastrophic shock, a corrective replacement is performed.The time to perform a minimal repair or a corrective replacement is negligible.External shocks and internal defects are independent.

(3) The system is replaced by a new one with negligible replacement time at aconstant time T (T > 0) after its installation, at a catastrophic shock or atdegradation failure whichever occurs first. After a replacement, the system isas good as new. The replacement time is negligible.

(4) The cost of a minimal repair is Cmr. The cost of a planned replacement at timeT is Cp. The cost of an unplanned replacement (due to a catastrophic shock ordue to a degradation failure) is Cc. All costs are positive numbers and verifyCc > Cp > Cmr.

The problem is to determine an optimal replacement time T . The optimizationproblem is formulated in terms of the expected cost rate given by (1) to be minimizedwith respect to T .

Let S1, S2, . . . , Sn, . . . be the arrival times of the defects. Since {Nd(t), t ≥ 0}follows a NHPP with intensity m(t), it follows from Kuniewski et al. (2009) that the

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conditional joint probability density function of (S1, S2, . . . , Sn) given {Nd(t) = n}for n = 1, 2, . . . is

fS1,S2,...,Sn(s1, s2, . . . , sn |n) =n!∏n

i=1 m(si)M(t)n

, 0 < s1 < s2 · · · < sn ≤ t, (3)

where M(t) is given by (2). Taking into account (3), one has

P [Nd(t) = n] = exp{−M(t)}∫ t

0

m(s1)ds1

∫ t

s1

m(s2)ds2 . . .

∫ t

sn−1

m(sn)dsn,

t ≥ 0, (4)

where s0 = 0 and n = 1, 2, . . . . Notice that, expression (4) is equivalent to

P [Nd(t) = n] = exp{−M(t)}M(t)n

n!, n = 1, . . . ,

but (4) considers the arrival times of the defects. Given that a defect has occurred,its growth starts and follows a gamma process. Let X i

t , i = 1, 2, . . . be the deterio-ration of the ith defect t units of time after its initiation. We assume X i

t follows agamma distribution with shape and scale parameters given by αt and β respectively(α, β > 0). It means that, for s < t, the density of the increment X i

t −X is of the ith

defect is given by

fα(t−s),β(x) =βα(t−s)

Γ(α(t − s))xα(t−s)−1e−βx, x ≥ 0, (5)

for i = 1, 2, . . . where Γ denotes the gamma function given by

Γ(α) =∫ ∞

0

uα−1e−udu.

A defect causes the system failure if its deterioration level exceeds a failure thresholdL. Let Y be the first time at which the deterioration level of a defect exceeds thefailure threshold L from the defect initiation. This distribution is given by

FY (t) = P [X it ≥ L] =

∫ ∞

L

fαt,β(x)dx =Γ(αt, Lβ)

Γ(αt), t ≥ 0, (6)

where fαt,β(x) is given by (5) and

Γ(α, x) =∫ ∞

x

zα−1e−zdz,

denotes the incomplete gamma function for x ≥ 0 and α > 0.External shocks arrive to the system according to a NHPP with intensity h(t).

They are minimally repaired with probability p and catastrophic with probability1 − p. Let {Nr,s(t), t ≥ 0} and {Nnr,s(t), t ≥ 0} be the number of repairable andcatastrophic shocks respectively in [0, t]. Then {Nr,s(t), t ≥ 0} and {Nnr,s(t), t ≥ 0}are NHPPs with intensity functions ph(t) and (1 − p)h(t) respectively.

Consider a replacement cycle defined by the time interval between successivereplacements of the system caused by a degradation failure, by a catastrophic shock

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or by a planned replacement at time T . We denote by XT the time to the replace-ment of the system and by Xf the time to a nonrepairable failure (due to a degra-dation failure or due to a catastrophic shock). Under the assumptions of the model,XT = min(Xf , T ) and

E[XT ] =∫ T

0

Ff (t)dt, (7)

where Ff (t) denotes the survival function of Xf . Notice that

Ff (t) =∞∑

i=0

Pi(t), t ≥ 0, (8)

where Pi(t) denotes the probability of the event {Xf ≥ t, Nd(t) = i} with t ≥ 0and i = 0, 1, 2, . . . . Considering (4), the corresponding probabilities for t ≥ 0 aregiven by

P0(t) = exp{−∫ t

0

((1 − p)h(u) + m(u))du

},

Pi(t) = P0(t)∫ t

0

m(s1)∫ t

s1

m(s2) . . .

∫ t

si−1

m(si)

×i∏

n=1

FY (t − sn)ds1ds2 . . . dsi, i = 1, 2, . . . ,

where s0 = 0, FY (t) = 1 − FY (t) and FY is given by (6).Under this maintenance scheme, the expected number of minimal repairs in a

replacement cycle is

E[Nr,s(XT )] =∫ T

0

pH(t)dFf (t)dt + pH(T )Ff (T )

= p

∫ T

0

h(t)Ff (t)dt, (9)

where h(t) denotes the intensity of the occurrence of shocks and Ff is given by (8).Let C(T ) be the expected cost rate given by (1). Using (7) and (9), we obtain

C(T ) =CcFf (T ) + CpFf (T ) + Cmr

∫ T

0ph(t)Ff (t)dt∫ T

0Ff (t)dt

, T > 0, (10)

where Ff is given by (8).

3. Optimization

The problem is to find a value of T that minimizes C(T ) given by (10), in otherwords to find a value Topt such that

C(Topt) = inf{C(T ) : T ≥ 0}. (11)

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Expression (10) can be expressed in a general form as

C(T ) =Cp +

∫ T

0 a(t)Ff (t)dt∫ T

0 Ff (t)dt, T > 0, (12)

where a(t) is given by

a(t) = (Cc − Cp)hf (t) + Cmrph(t), t ≥ 0, (13)

and hf (t) denotes the failure rate function of Xf at time t. The minimizationproblem of C(T ) detailed in (11) is a special case of the framework studied in Aven(1992), pp. 151, 152. Assuming a(∞) = limT→∞ a(T ) exists, the following resultholds.

Proposition 1. Let C(T ) and a(t) be defined by (12) and (13) respectively.

(1) If a(t) is strictly increasing in t and a(∞) > C(∞), there exists an unique andfinite minimum Topt that minimizes the function C(T ).

(2) If a(t) is nondecreasing in t and a(∞) < C(∞) < ∞, then Topt = ∞.

Proof 1. The proof of the proposition is evident since limT→0 C(T ) = ∞ and

C′(T ) ≥ (=) ≤ 0 ⇔ a(T ) ≥ (=) ≤ C(T ).

From Proposition 1, the optimization problem showed in (11) can reduce to theanalysis of the monotonicity of a(t) given by (13). Since h(t) is strictly increasingin t, if hf (t) is strictly increasing in t, then a(t) has also this property. Hence, theoptimization problem can reduce to the analysis of the monotonicity of hf (t) thatwe analyze below.

Proposition 2. For i = 2, 3, . . . , let bi(t) be the function given by

bi(t) =∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−1

gt(si)dsi, t ≥ 0, (14)

where gt(s) is given by

gt(s) = m(s)FY (t − s), s ≤ t.

Then, b′i(t) = b′1(t)bi−1(t) where

b1(t) =∫ t

0

gt(s)ds, t ≥ 0. (15)

Proof 2. The derivative of bi(t) is given by

b′i(t) =∫ t

0

∂gt(s1)∂t

ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−2

gt(si−1)dsi−1

∫ t

si−1

gt(si)dsi

+∫ t

0

gt(s1)ds1

∫ t

s1

∂gt(s2)∂t

ds2 . . .

∫ t

si−2

gt(si−1)dsi−1

∫ t

si−1

gt(si)dsi

. . .

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+∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−2

∂gt(si−1)∂t

dsi−1

∫ t

si−1

gt(si)dsi

+∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−2

gt(si−1)dsi−1

×(

gt(t) +∫ t

si−1

∂gt(si)∂t

dsi

).

Notice that∫ t

0

∂gt(s1)∂t

ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−1

gt(si)dsi

=∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−2

gt(si−1)dsi−1

∫ s1

0

∂gt(si)∂t

dsi,

and for j = 2, 3, . . . i − 1∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

sj−1

∂gt(sj)∂t

dsj . . .

∫ t

si−2

gt(si−1)dsi−1

∫ t

si−1

gt(si)dsi

=∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−2

gt(si−1)dsi−1

∫ sj

sj−1

∂gt(si)∂t

dsi.

Hence,

b′i(t) =∫ t

0

gt(s1)ds1

∫ t

s1

gt(s2)ds2 . . .

∫ t

si−2

gt(si−1)dsi−1

(gt(t) +

∫ t

0

∂gt(si)∂t

dsi

),

and therefore

b′i(t) = b′1(t)bi−1(t), i = 2, 3, . . . ,

and the result holds.

Proposition 3. Under the assumptions of this maintenance model, the failure ratefunction hf(t) of the time to a nonrepairable failure is nondecreasing in t.

Proof 3. Let Ff (t) be the survival function of the time to a nonrepairable failureXf given by (8)

Ff (t) = exp{−∫ t

0

((1 − p)h(u) + m(u))du

}(1 +

∞∑n=1

bn(t)

), t ≥ 0,

where b1(t) and bn(t) for n = 2, 3, . . . are given by (15) and (14) respectively. Thenthe failure rate function hf (t) is given by

hf (t) = (1 − p)h(t) + m(t) − (∑∞

n=1b′n(t))

(1 +∑∞

n=1bn(t)), t ≥ 0.

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Considering Proposition 2, hf(t) is reduced to

hf (t) = (1 − p)h(t) + m(t) − b′1(t), t ≥ 0.

Taking into account that

b1(t) =∫ t

0

m(u)FY (t − u)du,

then

hf (t) = (1 − p)h(t) +∫ t

0

m(u)fY (t − u)du, t ≥ 0,

or, equivalently,

hf (t) = (1 − p)h(t) +∫ t

0

fY (u)m(t − u)du, t ≥ 0,

and, assuming that the failure rate functions h(t) and m(t) are nondecreasing in t,the failure rate function hf (t) of the time to a nonrepairable failure is nondecreasingin t.

Considering Proposition 3, function a(t) given by (13) is nondecreasing in t.Using this fact, Theorem 1 solves analytically the optimization problem exposedin (11).

Theorem 1. Under the assumptions of the model and assuming that limt→∞ a(t) =a(∞) exists where a(t) is given by (13),

(1) If

a(∞)E[Xf ] >

∫ ∞

0

a(t)Ff (t)dt + Cp,

there exists an unique and finite value of Topt that minimizes the function C(T )given by (10). This value is obtained as the root of the equation C′(T ) = 0.

(2) If

a(∞)E[Xf ] <

∫ ∞

0

a(t)Ff (t)dt + Cp,

then Topt = ∞ and the optimal maintenance strategy is to replace the systemonly at failures.

4. Numerical Example

In this section, a numerical example is given to illustrate this maintenance model.We assume the defects arrive to the system following a homogeneous Poisson processof intensity m(t) = 0.002 defects per unit time. The deterioration of these defectsis modeled using gamma processes of parameters α = 0.004 and β = 1.5 (α/β

represents the average deterioration rate). The threshold failure is L = 10 units,that is, when the deterioration level of a defect exceeds the value 10, a degradation

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failure occurs. External shocks arrive according to a homogeneous Poisson process ofrate h(t) = µ = 0.001 shocks per time unit. With probability p = 0.95 these shocksare noncatastrophic and with probability 0.05 the shocks are catastrophic. The costsassociated to the different maintenance actions are Cp =10,000, Cc =100,000 andCmr = 500 monetary units.

Notice that, in this case,

a(t) = (Cc − Cp)hf (t) + Cmrph(t)

= 14.925 + 180FY (t),

hence limt→∞ a(t) exists and Theorem 1 can be applied.Figure 1 shows a simulation of the expected cost rate C(T ) versus T performed

using a MATLAB code with a finite differences procedure with 10,000 simulationsfor each point. By inspection, the value of T that minimizes C(T ) is Topt = 2,300units of time with an associated cost C(Topt) = 10.55 monetary units per unit time.

It is interesting to show how changes in the external factors affect to the optimalcost. For this task, different values for the frequency of external shocks are selected.These values vary from h(t) = µ = 0.001 to h(t) = µ = 0.002 shocks per timeunit. Figure 2 shows the optimal cost versus the frequency of external shocks. Asit is expected, the optimal cost increases when the frequency of shocks increases.Figure 3 shows the optimal cost versus the parameter β. When β increases, theaverage deterioration rate α/β decreases and the optimal cost decreases also.

We compare the strategy showed in this paper with the following strategy basedon the degradation measurement of the defects. Under the same framework showedin this paper (system subject to internal defects and external shocks), we assumethe system is preventively replaced when the degradation level of a defect exceeds

0 1000 2000 3000 4000 5000 60000

10

20

30

40

50

60

70

80

90

100

Preventive replacement time T(time units)

C(T

)(m

onet

ary

units

per

tim

e un

it)

Fig. 1. Expected cost rate as function of T .

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0.001 0.0012 0.0014 0.0016 0.0018 0.002 10

11

12

13

14

15

16

µ (shocks per unit time)

C(T

opt)

(mon

etar

y un

its p

er ti

me

unit)

Fig. 2. Optimal cost versus frequency of external shocks.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 28

9

10

11

12

13

14

β ({length unit}−1)

C(T

opt)

(mon

etar

y un

its p

er ti

me

unit)

Fig. 3. Optimal cost versus β.

a preventive maintenance threshold (namely M) and correctively replaced after acatastrophic failure. Noncatastrophic shocks are repaired using minimal repairs.The expected cost rate for this model is given by

C(M) =

Cp + (Cc − Cp)∫∞0 Fc(u)−d

du exp(−(1 − p)H(u))du

+ Cmr

∫∞0 ph(u)Fc(u) exp(−(1 − p)H(u))du∫∞

0 Fc(u) exp(−(1 − p)H(u))du, (16)

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I. T. Castro

for 0 ≤ M < L where

Fc(t) =∞∑

i=0

Pi,c(t), t ≥ 0, (17)

where Pi,c(t) are given by

P0,c(t) = exp{−∫ t

0

m(u)du

},

Pi,c(t) = P0,c(t)∫ t

0

m(s1)∫ t

s1

m(s2) . . .

∫ t

si−1

m(si)

×i∏

n=1

FσM (t − sn)ds1ds2 . . . dsi, i = 1, 2, . . . ,

where s0 = 0, FσM (t) = 1 − FσM (t) and FσM is given by

FσM (t) = P [X it ≥ M ] =

∫ ∞

M

fαt,β(x)dx =Γ(αt, Mβ)

Γ(αt), t ≥ 0.

Figure 4 shows a simulation of the expected cost rate C(M) given in (16) versusthe degradation level of the defects using the same data of the previous numericalexample. This simulation has been performed using a MATLAB code with a finitedifferences procedure with 10,000 simulations for each point. The objective costfunction C(M) is always nonincreasing in M and C(Mopt) = 7.7210 monetary unitsper unit time. From Fig. 4, the condition-based optimal cost is less than the age-based optimal cost. Although condition-based maintenance is considered as a more

1 2 3 4 5 6 7 8 9 107

8

9

10

11

12

13

14

15

16

Level deterioration M(length units)

C(M

)(m

onet

ary

units

per

uni

t tim

e)

Fig. 4. Expected cost rate versus M .

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advance approach, it is not universally superior to the age based policy (see examplesgiven by Pandey et al., (2009)). For example, setting cM the monitoring cost perunit time, the age-based maintenance policy is better (in terms of the expected costrate) than the condition-based maintenance policy if cM > C(Topt) − C(Mopt).

5. Conclusions

This work shows a general maintenance strategy for a system subject to multipleinternal defects and external shocks. An optimal preventive maintenance time T

that minimizes the expected cost rate is found analytically under the assumptionsof nondecreasing failure rates for the arrival of internal defects and external shocks.

Most papers that deal with competing DTS models assume that the degradationprocess and the shock process are independent. But, in many practical situations,the dependence between them is of importance and should not be neglected (Castroet al., 2010; Huynh et al., 2011). A natural extension of this paper is to introducethe dependence between shocks and defects.

Acknowledgments

This research was supported by the Ministerio de Educacion y Ciencia, Spain, undergrant MTM2012-36603-C02-01. The author is grateful to the anonymous reviewersfor useful comments and suggestions to an earlier version of the paper.

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Inmaculada T. Castro received her PhD degree in Mathematics from theUniversity of Granada, Spain in 2000. She worked at the Goverment of Andalu-cia as a research fellow for three years. She has published more than 20 articlesabout stochastic models in reliability in well-known journals including Advances inApplied Probability, Applied Stochastic Models in Business and Industry, ReliabilityEngineering and System Safety, Journal of Applied Probability and European Jour-nal of Operational Research. She currently holds a position of Associate Professorat the Department of Mathematics in the University of Extremadura, Spain.

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Documents

AN AGE-BASED MAINTENANCE STRATEGY FOR A DEGRADATION-THRESHOLD-SHOCK-MODEL FOR A SYSTEM SUBJECTED TO MULTIPLE DEFECTS