Optimal (τ, T) opportunistic maintenance of a k-out-of-n:G system with imperfect PM and partial failure

Optimal (τ, T ) Opportunistic Maintenance of a k-out-of-n:GSystem with Imperfect PM and Partial Failure

Hoang Pham, Hongzhou Wang∗

Department of Industrial Engineering, Rutgers University,Piscataway, New Jersey 08854-0909

Received December 1996; revised September 1998; accepted 21 October 1999

Abstract: The opportunistic maintenance of a k-out-of-n:G system with imperfect preventivemaintenance (PM) is studied in this paper, where partial failure is allowed. In many applications, theoptimal maintenance actions for one component often depend on the states of the other componentsand system reliability requirements. Two new (τ, T ) opportunistic maintenance models with theconsideration of reliability requirements are proposed. In these two models, only minimal repairsare performed on failed components before time τ and the corrective maintenance (CM) of allfailed components are combined with PM of all functioning but deteriorated components afterτ ; if the system survives to time T without perfect maintenance, it will be subject to PM at timeT . Considering maintenance time, asymptotic system cost rate and availability are derived. Theresults obtained generalize and unify some previous research in this area. Application to aircraftengine maintenance is presented. c© 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 223–239,2000

Keywords: opportunistic maintenance; imperfect maintenance; k-out-of-n:G system; cost rate;availability; partial failure; optimization

1. INTRODUCTION

For a complex and expensive system, it may not be advisable to replace the entire systemjust because of the failure of one component. In fact, the system comes back into operation onrepair or replacement of the failed component by a new one or by a used but operative one. Suchmaintenance actions do not renew the system completely but enable the system to continue tooperate [6, 11]. However, the system is usually deteriorating with usage and time. At some pointof time or usage, it may be in a bad operating condition and a perfect maintenance is necessary.Based on this situation, the following maintenance policy for a k-out-of-n system is designed.

A new system starts to operate at time 0. Each failure of a component of this system in thetime interval (0, τ) is immediately removed by a minimal repair. Components which fail in thetime interval (τ, T ) can be lying idle (partial failure is allowed). Perform corrective maintenance(CM) on the failed components together with preventive maintenance (PM) on all unfailed butdeteriorating ones at a cost of cf once exactly m components are idle, or perform PM on the wholesystem at a cost of cp once the total operating time reaches T , whichever occurs first. That is, if

∗Present address: Lucent Technologies, New Jersey.Correspondence to: H. Pham

c© 2000 John Wiley & Sons, Inc.

224 Naval Research Logistics, Vol. 47 (2000)

m components fail in the time interval (τ, T ), CM combined with PM is performed; if less thanm components fail in the time interval (τ, T ), then PM is carried out at time T . After a perfectmaintenance, either a CM combined with PM or a PM at T , the process repeats.

In the above maintenance policy, τ and T are decision variables. We assume that m is apredetermined positive integer, where 1 ≤ m ≤ n − k + 1 and real numbers τ < T . This policyis sketched in Figure 1. In practice, m may take different values according to different reliabilityand cost requirements. Obviously, m = 1 means that the system is subject to maintenancewhenever one component fails after τ . For a series system (n-out-of-n system) or a system withkey applications, m may basically be required to be 1. If m is chosen as n − k + 1, then thek-out-of-n system is maintained once the system fails. In most applications, the whole system issubject to a perfect CM together with a PM upon a system failure (m = n − k + 1) or partialfailure (m < n − k + 1). Here partial failure means some components have failed but the systemstill functions. However, if inspection is not continuous and the system operating condition canbe known only through inspection, m could be a number greater than (n− k +1). For simplicity,in this paper we assume that if CM together with PM is carried out both are perfect, given thatCM combined with PM takes w1 time units and PM at time T takes w2 time units. We furthersuppose that every component has increasing failure rate (IFR), which is differentiable and remainsundisturbed by minimal repair.

The justification of this policy is: Before τ , each component is ‘‘young’’ and no major repairis necessary. Therefore, only minimal repairs, which may not take much time and money, areperformed. The component is deteriorating as the time passes. After τ , the component has a largerfailure rate (due to IFR) and might be in a bad operating condition. Thus, a major or perfect repairmay be needed. Because there exist economic dependence and availability requirements (lessfrequent shutoffs for maintenance), however, we may not replace failed components immediatelybut start CM until the number of failed components reaches some prespecified number m. Infact, when the number of failed components accumulates to m, the remaining (n − m) operatingcomponents may degrade to a worse operating condition and need PM also. Note that as long asm is less than (n − k + 1), the system will not fail and continue to operate.

Economic dependence here means that it spends less cost and time to perform maintenance onseveral components jointly than on each component separately. For a multicomponent system, ifthere exists strong economic dependency, joint maintenance should be considered. The optimalmaintenance policy for this kind of systems possesses an opportunistic characteristic, i.e., theoptimal maintenance actions for one component depend on the states of the other components[21]. Obviously, the maintenance policy proposed above is an opportunistic one.

Economic dependency is common in most continuous operating systems. Examples of this typeof systems include aircraft, ship, power plants, telecommunication systems, chemical processingfacilities, and mass-production manufacturing lines while on a mission. For this type of systems,the cost of system unavailability (one-time shutdown) may be much higher than component main-tenance costs. Thus, there are often great potential cost savings by implementing an opportunisticmaintenance policy [5, 10] to reduce the frequency of shutdown maintenances.

Figure 1. Opportunistic maintenance policy.

Pham and Wang: Opportunistic Maintenance 225

In this paper, a k-out-of-n system:G is defined as a complex coherent system with n failure-independent components such that the system operates if and only if at least k of these componentsfunction successfully. Throughout this study, maintenance will be a general term and may representPM or CM. Replacement is a perfect maintenance. Repair is an action made at component or systemfailure and has the same meaning as CM. The terms CM and repair will be used alternativelywhen no confuse.

In this paper, the following supplementary assumptions are made:

I. All failure events are s-independent.II. Minimal repair takes negligible time since minimal repair time is small relatively

to perfect maintenance time.III. Minimal repair costs are random variables which depend on system age and

number of minimal repairs.IV. The planning horizon is infinite.V. k-out-of-n system consists of n statistically independent and identically dis-

tributed (i.i.d.) components.

This paper assumes that for each component the cost of the ith minimal repair at age t consists oftwo parts: the deterministic part c1(t, i), which depends on the age t of this component and thenumber i of minimal repairs, and the age-dependent random part c2(t). This general cost structurewas used by Sheu [13] and Sheu and Liou [14].

The supposition that the failure rate of each component is IFR is necessary. This is becausethe system may be subject to a PM at time T , which requires the system to be IFR after τ . Thefollowing proposition state the relationship between component and system failure rates:

PROPOSITION 1: If a k-out-of-n system is composed of independent, identical, IFR compo-nents, the system has an IFR also.

The proof of this proposition refers to Barlow and Proschan [2], which contains a similar propo-sition.

2. NOTATION

cf = cost of CM combined with PM of all unfailed but deteriorating components,cp = cost of PM alone of a system,

g(c1(t, i), c2(t)) = cost of the ith minimal repair at age t, where g is a positive, nondecreasingand continuous function,

c1(t, i) = deterministic part of cost of the ith minimal repair at age t,c2(t) = random part of cost of the ith minimal repair at age t,Vt(x) = cumulative distribution function of c2(t),vt(x) = probability density function of c2(t),τ, T = two constants in the (τ, T ) policy,f(t) = probability density function of a component,F (t) = cumulative density function of a component,F (t) = survival function of a component, F (t) = 1 − F (t),G(t) = residual survival function of a component,

Fn−k+1(y) = survival function of the time to failure of a k-out-of-n system,q(t) = failure rate of a component,Q(t) = cumulative failure rate of a component, Q(t) =

∫ t

0 q(x) dx,


n = number of components in a system,k = minimum number of operating components to make a system function,m = minimum number of failed components needed to start maintenance,w1 = time to perform CM together with PM,w2 = time to perform PM alone,p = probability that PM is perfect,q = q = 1 − p,

N(t) = number of minimal repairs during time interval (0, t),M(t) = expected number of minimal repairs during time interval (0, t),

L(τ, T ) = long-run expected system maintenance cost per unit time, or cost rate.

3. PERFECT PM

This section assumes that PM at time T is perfect. To facilitate the derivation of this mainte-nance model, we shall first characterize the classes of possible maintenance actions. From themaintenance policy described in Section 1, it is easy to see that, at any instant of time, the followingalternative maintenance actions for the k-out-of-n:G system are to be performed:

I. Keep the present system and no maintenance actions are given.II. Performed minimal repair on a component of the system (before time τ ).

III. Performed perfect repair on all failed components combined with PM on allunfailed but deteriorating components (after time τ ).

IV. Performed PM on the system at time T .

Given that each component in a k-out-of-n:G system has cumulative distribution function (cdf )F (x) and probability density function (pdf) f(x), its failure rate (or the hazard rate) is q(x) =f(x)/F (x). Its cumulative hazard is Q(x) =

∫ x

0 q(t) dt, which has a relationship with its survivalfunction F (x) = exp−Q(x), where F (x) = 1−F (x). If no PM, the residual survival functionof each component is given by

G(y) = PY ≥ τ + y|Y > τ

=∫ +∞

τ+y

f(t) dt/∫ +∞

τ

f(t) dt

= F (τ + y)/F (τ)

= e−Q(τ+y)+Q(τ), (1)

where y ≥ 0.Let Y1, Y2, . . . , Yn be i.i.d. random variables with survival distribution G(y), and Y(1) ≤ Y(2) ≤

· · · ≤ Y(n) be the corresponding order statistics. Note that the order statistics may be interpretedas successive times of failures of components in the systems, and the (n−k+1)th-order statistic isjust the time to failure of the k-out-of-n system. The order statistic Y(j) has the survival distribution

Fj(y) =j−1∑i=0

(ni

)[1 − G(y)]i[G(y)]n−i

=j−1∑i=0

(ni

)[1 − e−Q(τ+y)+Q(τ)]ie−(n−i)·Q(τ+y)+(n−i)·Q(τ), j = 1, 2, . . . , n. (2)


According to renewal theory of stochastic processes, the times between consecutive perfect main-tenance, preventive or corrective, constitute a renewal cycle. From the classical renewal rewardtheory, the long-run expected system maintenance cost per unit time, or cost rate, is

L(τ, T ) =C(τ, T )D(τ, T )

,

where C(τ, T ) is the expected system maintenance cost per renewal cycle and D(τ, T ) is theexpected duration of a renewal cycle.

Let Z1, Z2, . . . be i.i.d. random variables with cdf Fm(y) = 1−Fm(y), and Z∗i = min(Zi, T −

τ) for i = 1, 2, . . .. Note that we assume that PM at time T is perfect in this section. Therefore, arenewal cycle consists of maintenance time and Z∗

i duration. It is easy to verify:

D(τ, T ) = E[Z∗i ] + w1 · I[0,T−τ)(Zm) + w2 · I[T−τ,∞)(Zm),

= τ +∫ T−τ

0Fm(t) dt + w1 · I[0,T−τ)(Zm) + w2 · I[T−τ,∞)(Zm)

= τ +∫ T−τ

0Fm(t) dt + Fm(T − τ)(w1 − w2) + w2. (3)

Next we evaluate expected system maintenance cost per renewal cycle C(τ, T ). Note thatC(τ, T ) consists of three parts: minimal repair cost, cost of CM combined with PM, and costof PM at time T . For each component the failures between (0, τ) occur in accordance with anonhomogeneous Poisson process (NHPP) of rate q(t) [1, 3, 4]. The cost of the ith minimal repairat age t is g(c1(t, i), c2(t)), where g is a positive, nondecreasing and continuous function of t, andis a positive, nondecreasing function of i. Given that the random part c2(t) at age t has distributionfunction Vt(x), density function vt(x), and finite mean E[c2(t)], the total minimal repair cost fora k-out-of-n system in one renewal cycle is given by

Csmr = nE

N(τ)∑

i=1

g(c1(Si, i), c2(Si))

,

where N(τ) is the number of minimal repairs of each component during time interval (0, τ),and E[

∑N(τ)i=1 g(c1(Si, i), c2(Si))] is the expected total minimal repair cost of each component

in (0, τ).The further derivation of this cost expression needs a proposition from Sheu [13], and we now

state it without proof:

PROPOSITION 2: Let N(t), t ≥ 0 be a nonhomogeneous Poisson process with intensityq(t) and M(t) = E[N(t)] =

∫ t

0 q(u) du. Denote the successive arrival times of this process byS1, S2, . . .. Assume that at time Si a cost of g(c1(Si, i), c2(Si)) is incurred. Suppose that c2(y)at age y is a random variable with finite mean E[c2(t)] and g is a positive, nondecreasing, andcontinuous function. If A(t) is the total cost incurred over [0, t], then

E[A(t)] =∫ t

0µ(y)q(y) dy,


where µ(y) = EN(y)bEC2(y)[g(c1(y, N(y) + 1), c2(y))]c, which is the expectation with respectto random variables N(y) and c2(y).

Note that Si in the above proposition corresponds to the ith failure of each component in timeinterval (0, τ) at which a minimal repair cost g(c1(Si, i), c2(Si)) is induced. Given that the failuresof each component occur in accordance with a NHPP of intensity q(t), the total minimal repaircost in one cycle is, from this proposition, given by

Csmr = n

∫ τ

0µ(y)q(y) dy, (4)

where µ(y) = EN(y)bEC2(y)[g(c1(y, N(y) + 1), c2(y))]c.Besides, it is easy to ascertain the total maintenance cost of CM together with PM and PM

alone:

Cpf = cf · I[0,T−τ)(Zm) + cp · I[T−τ,∞)(Zm)

= Fm(T − τ)(cf − cp) + cp.

Thus,

C(τ, T ) = Csmr + Cpf

= n

∫ τ

0µ(y)q(y) dy + Fm(T − τ)(cf − cp) + cp. (5)

From Eqs. (3) and (5), the following proposition follows:

PROPOSITION 3: For the opportunistic maintenance policy described in Section 1, if PM isalways perfect, then the long-run expected system maintenance cost per unit time, or cost rate,for a k-out-of-n system:G is given by

L(τ, T ) =n

∫ τ

0 µ(y)q(y) dy + Fm(T − τ)(cf − cp) + cp

τ +∫ T−τ

0 Fm(t) dt + Fm(T − τ)(w1 − w2) + w2

, (6)

and the limiting average system availability is

A(τ, T ) =τ +

∫ T−τ

0 Fm(t) dt

τ +∫ T−τ

0 Fm(t) dt + Fm(T − τ)(w1 − w2) + w2

. (6′)

In what follows, we shall attempt to minimize L(τ, T ) with respect to (τ, T ). DifferentiatingL(τ, T ) with respect to T and τ , respectively, we have

∂L(τ, T )∂T

= − Fm(T − τ)[n∫ τ

0 µ(y)q(y) dy + (cf − cp) · Fm(T − τ) + cp]

[τ +∫ T−τ

0 Fm(t) dt + Fm(T − τ)(w1 − w2) + w2]2

+

∂Fm(T − τ)∂T

[τ(cf − cp) + cfw2 − cpw1 + (cf − cp)


×∫ T−τ

0Fm(t) dt − n(w1 − w2)

∫ τ

0µ(y)q(y) dy

]

×[τ +

∫ T−τ

0Fm(t) dt + Fm(T − τ)(w1 − w2) + w2

]−2

,

∂L(τ, T )∂τ

=

[nµ(τ)q(τ) + ∂Fm(T−τ)

∂τ (cf − cp)]

× [τ +∫ T−τ

0 Fm(t) dt + Fm(T − τ)(w1 − w2) + w2]

[τ +∫ T−τ

0 Fm(t) dt + Fm(T − τ)(w1 − w2) + w2]2

−[n

∫ τ

0µ(y)q(y) dy + Fm(T − τ)(cf − cp) + cp

]

×[1 +

∫ T−τ

0

∂Fm(t)∂t

dt − Fm(T − τ) +∂Fm(T − τ)

∂τ(w1 − w2)

]

×[τ +

∫ T−τ

0Fm(t) dt + Fm(T − τ)(w1 − w2) + w2

]−2

.

A necessary condition that a pair (τ∗, T ∗) minimizes L(τ, T ) is that it satisfies

∂L(τ, T )∂τ

= 0 and∂L(τ, T )

∂T= 0

The optimal (τ, T ) maintenance policy is obtained by solving the above equations.

4. IMPERFECT PM

In Section 3 we assume that PM is always perfect. In practice, this assumption may not berealistic in some cases. This section assumes that PM is imperfect, i.e., this section is the same asSection 3 except that in this section PM at time T, 2T, 3T, . . ., is imperfect. Generally, imperfectmaintenance refers to any maintenance which makes a unit not ‘‘as good as new’’ but younger.Usually, it is assumed that imperfect maintenance restores the unit’s operating state to somewherebetween ‘‘as good as new’’ and ‘‘as bad as old.’’ Clearly, imperfect maintenance is a generalmaintenance which can include two extreme cases: minimal and perfect maintenance. An overviewof imperfect maintenance is presented in [12]. In this paper, imperfect PM is treated in a way thatafter PM a k-out-of-n system is good as new with probability p (perfect PM) and is bad as oldwith probability q = 1−p (minimal PM, 0 ≤ p ≤ 1) [8, 12, 17]. Details on this treatment methodfor imperfect maintenance can be seen in [12]. Other assumptions and notations are identical tothose in Section 3.

According to renewal theory of stochastic processes, the times between consecutive perfectmaintenance, preventive or corrective, constitute a renewal cycle. From the classical renewalreward theory, the long-run expected system maintenance cost per unit time, or cost rate withparameter p, is

L(τ, T | p) =C(τ, T | p)D(τ, T | p)

,


where C(τ, T | p) is the expected system maintenance cost per renewal cycle and D(τ, T | p) isthe expected duration of a renewal cycle.

LetZ1, Z2, . . .be i.i.d. random variables with survival function Fm(y), andZ∗i = min(Zi, kT−

τ | k = number of PM until the first perfect one occurs) for k, i = 1, 2, . . .. Similar to last section,a renewal cycle consists of maintenance time and Z∗

i duration, Thus

D(τ, T | p) = E[Z∗i ] + expected maintenance time. (7a)

Let Tp be the first perfect PM-alone time point. Note that Tp = T, Tp = 2T, Tp = 3T, . . .,are mutually disjoint events satisfying sample space Ω = ∪∞

j=1Tp = jT. Note that a renewalcycle is completed either by any CM combined with PM or by a perfect PM at time kT . Noticealso that the probability that a PM alone is perfect is p. It follows that

E[Z∗i ] = E[Z∗

i | Tp = T ] · I[T ](Tp) + E[Z∗i | Tp = 2T ] · I[2T ](Tp) + · · ·

= p

[τ +

∫ T−τ

0Fm(t) dt

]+ qp

[τ +

∫ 2T−τ

0Fm(t) dt

]

+ q2p

[τ +

∫ 3T−τ

0Fm(t) dt

]+ · · ·

= τ · [p + qp + q2p + · · ·]

+ p

∫ T−τ

0+qp

∫ T−τ

0+qp

∫ 2T−τ

T−τ

+ q2p

∫ T−τ

0+q2p

∫ 2T−τ

T−τ

+q2p

∫ 3T−τ

2T−τ

+ · · ·

= τ + p(1 + q + q2 + · · ·)∫ T−τ

0+pq(1 + q + q2 + · · ·)

∫ 2T−τ

T−τ

+ · · ·

= τ + p · 11 − q

∫ T−τ

0+q

∫ 2T−τ

T−τ

+q2∫ 3T−τ

2T−τ

+ · · ·

= τ +∫ T−τ

0Fm(t) dt +

∞∑j=2

qj−1∫ jT−τ

(j−1)T−τ

Fm(t) dt. (7b)

Let CM be the event that CM together with PM is performed in a renewal cycle. From the Theoremof Total Probabilities, the probability that CM combined with PM appears is

P (CM) = P (CM | Tp = T ) · I[T ](Tp) + P (CM | Tp = 2T ) · I[2T ](Tp) + · · ·= p · Fm(T − τ) + qp · Fm(2T − τ) + q2p · Fm(3T − τ) + · · ·

= p

∞∑j=1

qj−1Fm(jT − τ)


= p

1

1 − q−

∞∑j=1

qj−1Fm(jT − τ)

= 1 − p∞∑

j=1

qj−1Fm(jT − τ). (7c)

Similarly, we can obtain the probability P (PM) that PM alone occurs which is

∞∑j=1

qj−1Fm(jT − τ). (7d)

The above expression (7d) also has direct meaning. For example, q · Fm(2T − τ) represents theprobability that less than m components have failed in the interval (τ, 2T ) and the first PM turnsout to be not perfect (with probability q). Obviously,

expected maintenance time = w1 · P (CM) + w2 · P (PM). (7e)

It follows from Eqs. (7a)–(7e) that

D(τ, T | p) = E[Z∗i ] + expected maintenance time

= τ +∫ T−τ

0Fm(t) dt +

∞∑j=2

qj−1∫ jT−τ

(j−1)T−τ

Fm(t) dt

+w1

1 − p

∞∑j=1

qj−1Fm(jT − τ)

+ w2

∞∑

j=1

qj−1Fm(jT − τ)

. (7′)

Next we determine expected system maintenance cost per renewal cycle C(τ, T | p), whichconsists of three parts: minimal repair cost, PM cost, and cost of CM together with PM. The totalminimal repair cost in one cycle is the same as the one in Eq. (4). Again note that a renewal cycleis completed either by any CM together with PM or by a perfect PM, and that the probability thata PM alone is perfect is p. Similarly to the derivation of the expected maintenance time, it is easyto show that the total cost of PM alone and CM combined with PM is given by

Cpf = cp

∞∑j=1

qj−1Fm(jT − τ) + cf

1 − p

∞∑j=1

qj−1Fm(jT − τ)

.

Thus,

C(τ, T | p) = Csmr + Cpf

= n

∫ τ

0µ(y)q(y) dy + cp

∞∑j=1

qj−1Fm(jT − τ)

+ cf

1 − p

∞∑j=1

qj−1Fm(jT − τ)

. (8)

From Eqs. (7′) and (8) the following proposition follows:


PROPOSITION 4: For the opportunistic maintenance policy described in Section 1, if the PMis perfect with probability p and minimal with probability q = 1 − p, then the long-run expectedsystem maintenance cost per unit time, or cost rate, for a k-out-of-n system:G is given by

L(τ, T | p) =

n

∫ τ

0µ(y)q(y) dy + cp

∞∑j=1

qj−1Fm(jT − τ)

+ cf

1 − p

∞∑j=1

qj−1Fm(jT − τ)

×τ +

∫ T−τ

0Fm(t) dt +

∞∑j=2

qj−1∫ jT−τ

(j−1)T−τ

Fm(t) dt

+ w1

1 − p

∞∑j=1

qj−1Fm(jT − τ)

+ w2

∞∑

j=1

qj−1Fm(jT − τ)

−1

, (9)

and the limiting average system availability is

A(τ, T | p) =

τ +

∫ T−τ

0Fm(t) dt +

∞∑j=2

qj−1∫ jT−τ

(j−1)T−τ

Fm(t) dt

×τ +

∫ T−τ

0Fm(t) dt +

∞∑j=2

qj−1∫ jT−τ

(j−1)T−τ

Fm(t) dt

+w1

1 − p

∞∑j=1

qj−1Fm(jT − τ)

+ w2

∞∑

j=1

qj−1Fm(jT − τ)

−1

. (9′)

Obviously, if we set p = 1 in Eqs. (9) and (9′) we will obtain Eqs. (6) and (6′). The optimal (τ, T )maintenance policy with parameter p can be obtained by the same method as in Section 3.

5. SPECIAL CASES

The two models in Sections 3 and 4 include some previous maintenance models as specialcases. A summary is given below. Since Proposition 3 is a special case of Proposition 4, the


discussions in this section will be focused on Proposition 3. The following special cases are interms of Eqs. (6) and (6′) except case 10.

CASE 1 (n = k = m = 1, w1 = w2 = 0, τ = 0): This is the classical age-replacement pol-icy which was called policy I in Barlow and Hunter [1]. If we set n = k = m = 1, w1 = w2 = 0,and τ = 0 in Eq. (6), then we will obtain the well-known result by Barlow and Hunter [1]:

L(0, T ) =F (T )cf + cpF (T )∫ T

0 F (t) dt

CASE 2 (m = n − k + 1, w1 = w2 = 0, τ = 0, G(y) = e−λy): Nakagawa [9] investigatedthis case. If we set m = n − k + 1, w1 = w2 = 0, τ = 0, and G(y) = e−λy in Eq. (6),then the cost rate becomes

L(0, T ) =cp + (cf − cp)

∑k−1i=0

(ni

)e−iλT [1 − e−λT ]n−i

∫ T

0

∑ni=k

(ni

)e−iλt[1 − e−λt]n−i dt

,

which agrees with Eq. (8) in Nakagawa [9].

CASE 3 (n = k = m = 1, w1 = w2 = 0, τ = T, g(c1(t, i), c2(t)) = c): This is policy II dis-cussed by Barlow and Hunter [1], that is, the classical periodic replacement with minimal repairat failure. If we set n = k = m = 1, w1 = w2 = 0, τ = T , and g(c1(t, i), c2(t)) = c in Eq. (6)the system maintenance cost rate becomes

L(T, T ) =cQ(T ) + cp

T,

which coincides with the well-known result by Barlow and Hunter [1].

CASE 4 (n = k = m = 1, w1 = w2 = 0, τ = T, g(c1(t, i), c2(t)) = c, cp = c(T )): This isthe case treated by Tilquin and Cleroux [16]. If we set n = k = m = 1, w1 = w2 = 0, τ =T, g(c1(t, i), c2(t)) = c, and cp = c0 + a(T ) in Eq. (6), then the system maintenance cost rate isgiven by

L(T, T ) =cQ(T ) + c0 + a(T )

T,

which is the same as the cost rate in Tilquin and Cleroux [16].

CASE 5 (n = k = m = 1, w1 = w2 = 0, τ = T, g(c1(t, i), c2(t)) = c(y)): This is the caseinvestigated by Boland [3].

CASE 6 (n = k = m = 1, w1 = w2 = 0, τ = T , g(c1(t, i), c2(t)) = ci): Boland and Pro-schan [4] studied this case. In particular, they considered the cost structure ci = a + ic inwhich minimal repair cost is increasing with the number of minimal repairs.

CASE 7 (n = k = m = 1, w1 = w2 = 0, g(c1(t, i), c2(t)) = c): This is the policy consid-ered by Tahara and Nishida [15]. If we set n = k = m = 1, w1 = w2 = 0, and g(c1(t, i), c2(t)) =c in Eq. (6), then the expected systems maintenance cost rate is

L(τ, T ) =cQ(τ) + G(T − τ)(cf − cp) + cp

τ +∫ T−τ

0 G(t) dt,


which agrees with Eq. (23) in Tahara and Nishida [15].

It is noted that Tahara and Nishida [15] discussed the optimality of the (τ, T ) policy for a one-unitsystem by means of dynamic programming techniques and showed that the (τ, T ) maintenancepolicy is optimal.

CASE 8 (n = k = m = 1, w1 = w2 = 0, T = ∞, g(c1(t, i), c2(t)) = c): Muth [7] studiedthis case. If we set n = k = m = 1, w1 = w2 = 0, T = ∞, and g(c1(t, i), c2(t)) = c inEq. (6), we will obtain the same result as in Muth [7]:

L(τ, T ) =cQ(τ) + cf

τ +∫ ∞0 G(t) dt

.

CASE 9 (n = k = m = 1, w1 = w2 = 0, T = ∞, g(c1(t, i), c2(t)) = c(t)): Yun [19] consid-ered this case. If we set n = k = m = 1, w1 = w2 = 0, T = ∞, and g(c1(t, i), c2(t)) = c(t) inEq. (6), we will get the same result as in Yun [19]:

L(τ, ∞) =

∫ τ

0 c(t)q(y) dy + cf

τ +∫ ∞0 e−Q(τ+x)+Q(τ) dt

.

CASE 10 (n = k = m = 1, w1 = w2 = 0, τ = 0) for Eq. (9): Nakagawa [8] investigated thiscase. If we set n = k = m = 1, w1 = w2 = 0, and τ = 0 in Eq. (9), then the cost rate becomes

L(0, T | p) =cp

∑∞j=1 qj−1F (jT ) + cfb1 − p

∑∞j=1 qj−1F (jT )c∑∞

j=1 qj−1∫ jT

(j−1)T F (t) dt,

which coincides with Eq. (1) in Nakagawa [8].

CASE 11 (n = k): This case corresponds to optimal (τ, T ) maintenance policy of a seriessystem. If we set n = k and m = 1 in Eq. (6), it follows that the long-run expected systemmaintenance cost rate for a series system with n component is

L(τ, T ) =n

∫ τ

0 µ(y)q(y) dy + [1 − Fn(T )/F−n(τ)](cf − cp) + cp

τ + [F (τ)]−n∫ T−τ

0 [F (τ + t)]n dt + [1 − Fn(T )/F−n(τ)](w1 − w2) + w2

.

CASE 12 (k = 1, n > 1): In this case the k-out-of-n system is reduced to a parallel system.If we let k = 1 and m = n, it follows that the long-run expected system maintenance cost ratefor a parallel system with n components is

L(τ, T ) =n

∫ τ

0 µ(y)q(y) dy + (cf − cp)[F (τ) − F (T )]nF−n(τ) + cp

τ +∫ T−τ

0 1 − [F (τ) − F (τ + t)]nF−n(τ) dt

+ (w1 − w2)[F (τ) − F (T )]nF−n(τ) + w2

.

If we set τ = 0 and w1 = w2 = 0, then the above equation becomes

L(0, T ) =(cf − cp)Fn(T ) + cp∫ T

0 [1 − Fn(t)] dt,


which coincides with the result in Yasui, Nakagawa, and Osaki [18].

6. OPTIMIZATION PROBLEMS

In Sections 3 and 4 we evaluate the expected system maintenance cost rate and availability.Sometimes the optimal maintenance policies may be required that the maintenance cost rate isminimized while some availability requirements are satisfied, or the system availability is maxi-mized given that maintenance cost rate is not larger than some predetermined value. For example,for maintenance model in Section 4, the following optimization problem can be formulated interms of decision variables T and τ :

Minimize

L(τ, T | p) =

n

∫ τ

0µ(y)q(y) dy + cp

∞∑j=1

qj−1Fm(jT − τ)

+ cf

1 − p

∞∑j=1

qj−1Fm(jT − τ)

×τ +

∫ T−τ

0Fm(t) dt +

∞∑j=2

qj−1∫ jT−τ

(j−1)T−τ

Fm(t) dt

+ w1

1 − p

∞∑j=1

qj−1Fm(jT − τ)

+ w2

∞∑

j=1

qj−1Fm(jT − τ)

−1

Subject to

A(τ, T | p) ≥ A0,τ ≥ 0,T > 0,

where constant A0 is the predecided minimum availability requirement.The optimal system maintenance policy (T ∗, τ∗) can be determined from the above optimiza-

tion model by using nonlinear programming software. Similarly, other optimization models canbe formulated based on different requirements. Such optimization models should be truly optimalsince both availability and maintenance costs are addressed.

7. APPLICATION EXAMPLE

This section discusses application to a 2-out-of-3 aircraft engine system. Suppose that time tofailure of each engine follows the Weibull distribution with shape parameter β and scale parameter


θ whose pdf is given by

f(y) =β

θ

(y

θ

)β−1exp

[−

(y

θ

)β]

, y > 0, β, θ > 0,

and failure rate q(y) = (β/θ)(y/θ)β−1. Assume that β = 2 and θ = 500 (days) and g(c1(y, i),c2(y)) = c1(y)+ c2(y). Since β = 2 > 1, the lifetime of each engine has IFR and by Proposition1 lifetime of this system has IFR. Assume that c1(y) = 1 +

√y and c2(y) follow the normal

distribution with mean 1. Then

µ(y) = EN(y)bEC2(y)[g(c1(y, N(y) + 1), c2(y))c= E[c1(y) + c2(y)]

= 2 +√

y.

Since the aircraft engine system is considered to be critical to the aircraft, m is taken to be 1 inthis example. Besides, let

w1 = 5 days, w2 = 4 days, cf = 60, cp = 40.

Substituting the above data into Eq. (6), it follows that

L(τ, T ) =6(τ2 + 2

5τ5/2)/5002 − 20 expb−3(0.002T )2 + 3(0.002τ)2c + 60

τ + 500 × 6−1/2 exp[3(0.002τ)2]∫ 0.002T

√6

0.002τ√

6 exp[− 12u2] du

− exp[−3(0.002T )2 + 3(0.002τ)2] + 5

.

Various kinds of approximations for the integral in the above equation have been developed anda simple approximation with high accuracy is by Zelen and Severo [20]:

∫ t

−∞exp( 1

2u2) du ≈ √2π[1− 1

2 (1+0.196854t− 0.115194t2 +0.000344t3 +0.019527t4)−4].

The error, for t ≥ 0, is less than 2.5 × 10−4.By nonlinear optimization software, the optimal solution is

τ∗ = 335.32 days, T ∗ = 383.99 days,

which results in the minimum system maintenance cost rate given by

L(τ∗, T ∗) = 0.1826.

That is, the optimal maintenance policy for this 2-out-of-3 aircraft engine system is that beforeτ∗ = 335.32 (days) only minimal repairs are performed; after τ∗ = 335.32: The failed enginewill be subject to perfect repair together with PM on the remaining two once any engine fails. Ifno engine fails until time T ∗ = 383.99 (days), PM is carried out at T ∗ = 383.99.


Now suppose that p = 0.90, A0 = 0.98, and other parameters are kept the same. The optimiza-tion model in Section 6 becomes

Minimize L(T, τ) = NN/DD,

Subject to

A(τ, T | p) = NN1/DD ≥ 0.98,τ ≥ 0,T > 0,

where

NN = 60 + 0.000024(τ2 + 0.4τ2.5) − 14 · expb0.000012(τ2 − T 2)c

− 14∞∑

j=2

0.1j−1 exp[0.000012(τ2 − (jT )2)],

NN1 = τ + 500 · 6−1/2 exp(0.000012τ2)

× ∞∑

j=2

0.1j−1∫ 0.002

√6jT

0.002√

6(j−1)Texp(−u2/2) du +

∫ 0.002√

6T

0.002√

6τ

exp(−u2/2) du

,

DD = 5 + τ + 500 · 6−1/2 exp(0.000012τ2)

× ∞∑

j=2

0.1j−1∫ 0.002

√6jT

0.002√

6(j−1)Texp(−u2/2) du +

∫ 0.002√

6T

0.002√

6τ

exp(−u2/2) du

− 0.5 exp[0.000012(τ2 − T 2)] − 0.5∞∑

j=2

0.1j−1 exp[0.000012(τ2 − (jT )2)].

The optimal solution for the above optimization model is

τ∗ = 316.89 days, T ∗ = 464.55 days

which generates the minimum system maintenance cost rate:

L(τ∗, T ∗) = 0.1868.

8. CONCLUDING REMARKS

In the two maintenance models developed in this paper, PM on nonfailed but degraded compo-nents is also carried out at the moment when CM activities are called for after τ . Such maintenancepolicies may reduce the number of unexpected CM activities at fairly low costs, since PM to-gether with CM can be performed without substantial additional expenses. Maintenance time isalso considered and partial failure is allowed in this paper. Besides, the following points shouldbe mentioned:


(1) Some group replacement policies do not contemplate system configuration and reliabilityrequirements. In fact, system configuration and reliability requirements are important to schedulemaintenance actions. For a k-out-of-n system, when one of its components is down if we repairor replace it there will be a one-time shutoff at which system will not be available. However, evenif we do not take any maintenance action on this component, the system may still operate as longas the number of failed components does not exceed n − k. Note that once the number of failedcomponents surpasses n − k the system fails. Thus, most maintenance actions may start earlierthan that moment.

(2) Equations (6), (6′), (9), and (9′) are still valid if CM and PM costs as well as maintenancetimes are random variables. In this case, cf , cp, w1, and w2 represent expected costs of CMcombined with PM, PM alone, expected maintenance time of CM combined with PM and PMalone, respectively, in Eqs. (6), (6′), (9), and (9′).

(3) m could be a decision variable according to different situations. Its optimal value, togetherwith the optimal values of τ and T , can be found by minimizing the system maintenance cost ratein Eqs. (6), (6′), (9), and (9′) and in Section 6 in terms of decision variables τ, T , and m.

(4) m could take a natural number greater than n − k + 1, depending on different reliabilityand maintenance cost requirements.

ACKNOWLEDGMENTS

The authors would like to thank an Associate Editor and two referees for their valuable commentsand suggestions, which have been already incorporated in the text.

REFERENCES

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Optimal (τ, T) opportunistic maintenance of a k-out-of-n:G system with imperfect PM and partial failure