A generalized Brown-Proschan model for preventive and [email protected] Jean Kuntzmann Laboratory Grenoble University France 1 of 28. ISBIS-2010 Laurent DOYEN

ISBIS-2010 Laurent DOYEN

A generalized Brown-Proschan model

for preventive and corrective maintenance

Laurent [email protected]

Jean Kuntzmann LaboratoryGrenoble University

France

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I. Introduction

The dependability of complex repairable systems depends strongly on theefficiency of maintenance actions.

Corrective Maintenance (CM): After a failure, put the system intoa state in which it can perform its function again.

Preventive Maintenance (PM): When the system is operating, in-tends to slow down the wear process.

Basic maintenance effect assumptions:

•As Bad As Old (ABAO): restores the system in the same state itwas just before maintenance.

• As Good As New (AGAN): restores the system as if it were new.

Reality is between these two extreme cases: Imperfect maintenance.2 of 28


Brown-Proschan model [83], CM is :

• AGAN with probability p,• ABAO with probability 1− p.• repair effects (AGAN or ABAO) are mutually independent

and independent of already observed failure times.

CM effects can be characterized with random variables:

Bi =

{1 if the ith CM is AGAN0 if the ith CM is ABAO

Statistical studies when the {Bi}i≥1 are known:Whitaker-Samaniego[89], Hollander-Presnell-Sethuraman [92], Kvam-Singh-Whitaker [02], Bathe-Franz [96], Agustin-Pena[99],...

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Practical purposes : {Bi}i≥0 are hidden variables.

p characterizes maintenance efficiency:

• p = 0 : ABAO

• p = 1 : AGAN

• 0 < p < 1 : imperfect

Joint assessment of CM efficiency and intrinsic wear-out:Lim [98]; Lim-Lie[00]; Lim-Lu-Park[98]; Langseth-Lindqvist [04].

Presentation aim:

• Generalize the BP model to PM effects

• Assess PM efficiency and intrinsic wear-out when PM effects areunknown

• Compute reliability indicators.

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II/ A maintenance data set

PM and CM times of a subsystem of a fossil-fired thermal plant of EDF:

1179 2640 4101 5562 7023 8035 8329 8376 8393 84558484 8494 8605 8628 8641 8744 8903 9105 9660 98459846 9866 9919 9985 9987 10010 10363 10470 11021 1149412851 12956 13662 14161 14217 14244

0 2000 4000 6000 8000 10000 12000 140000

5

10

15

20

25

Cumulative number of failures

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III. Notations

tt

t

tt

t

PM PMc CM CM CM TPM0

C1 C3 C4C2

u1 = 0 u2 = 0 u3 = 1 u4 = 0

τ1 τ2 τ3

T1 T2 T3

1

2

3

1

2

3

mt

Nt

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IV. Model assumptions

• λ(t) the failure rate of the new unmaintained system. Λ(t) =∫ t

0 λ(s) ds,

• Maintenance durations are not taken into account,

• PM are done at deterministic times, all the PM times are known

• CM are done at random times, CM are observed over [c, T ] (c < T )

• CM effects are ABAO,

• PM effects follow a Brown-Proschan model, i.e. PM effects are :

• mutually independent and independent of previous failure times,

• AGAN with probability p,

• ABAO with probability 1− p.Bi =

{1 if the ith PM is AGAN0 if the ith PM is ABAO

P (Bi = 1 | T i, Bi−1) = P (Bi = 1) = p

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V. Joint assessment of intrinsic wear-out and PM efficiencya/ Maximum likelihood method

Likelihood associated to a single observation of the failure process over[c, t]:

Lt(θ) = fTnt |Nt=nt(tnt)P (Nt = nt)

Maximum Likelihood Estimator (MLE):

θ = arg maxθ

LT (θ) = arg maxθ

log(LT (θ))

Notations: Lc(θ) = 1

Lτ,t(θ) = fTnt−nτ ,Nt−τ=nt−nτ (tnτ+1 − τ, ..., tnt − τ ) ⇒ the likelihood

associated to the system new in τ and observed over[c ∨ τ, t].

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Let us denote : Dmt = (1− p)mt−−m

∏c∨τm≤tj≤t

λ(tj − τm)

Two equivalent equations for likelihood recursive computation:

• Lt(θ) =

mt−∑m=0

p1l{m6=0}Dm

t e−(Λ(t−τm)−Λ((c∨τm)−τm))Lc∨τm(θ)

⇒ forward computation algorithm.

• Lt(θ) =

∑τ∈{τ1,...,τmt−

,t}p

1l{τ 6=t}D0τm e−(Λ(τm∨c)−Λ(c))Lτm,t(θ)

⇒ backward computation algorithm.

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Proof of the equation used for the forward computation:

{Bm = 1, {Bj = 0}m<j≤mt−︸︷︷︸τm is the last AGAN PM since t

}0≤m≤mt−is a partition of the probability space.

GivenBm = 1, the CM process can be divided into 2 independent processes.

Lt(θ) =[ mt−∑m=0

p1l{m6=0} (1− p)mt−−m︸︷︷︸

P (Bm=1,{Bj=0}m<j≤mt−

) ∏c∨τm≤tj≤t

λ(tj − τm)

e−(Λ(t−τm)−Λ((c∨τm)−τm))

︸︷︷︸Given Bm=1,{Bj=0}m<j≤m

t−, CM times after τm follow a NHPP initialized in τm

Lc∨τm(θ)]

︸︷︷︸Given Bm=1, maintenances before τm follow a BP PM-ABAO CM model

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b/ MLE combined with moment estimation : λ(t) = αβtβ−1

Power Law Process: E[NT ] = α(Tβ − cβ) ⇒ αβ = NT/(Tβ − cβ).

BP model , E[Nt] = αSt, where

St =∑

τ∈{τmc+1,...,τmt−,t}

mτ−∑k=0

p1l{k 6=0}(1− p)mτ−−k((τ − τk)β − ((τmτ−∨ c)− τk)β)

MLE combined with moment estimation of α: αβ,p = NTST

E[αβ,p] = α

(β, p) = arg max(β,p)

log(LT (αβ,p, β, p)) and α = αβ,p

⇒ Dimension reduction of the likelihood maximization space.

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c/ Individual PM efficiency assessment

p characterized the average global PM efficiency.

The mth PM effect can be characterized by:

πθm = P (Bm = 1 | NT = nT , TnT = tnT )

which verifies:

πθm = pLc∨τm(θ) Lτm,T (θ)

LT (θ)

It can naturally be estimated by: πθm

• Lτm(θ): intermediate computing values of the forward algorithm

• Lτm,T (θ): intermediate computing values of the backward algorithm12 of 28


d/ Expectation-Maximization algorithm

Complete likelihood:

Lct(θ) = fTnt |Nt=nt,Bnt=bnt(tnt)P (Nt = nt|Bnt = bnt)P (Bnt = bnt)

EM algorithm:

• Expectation (E) step: Compute Q(θ|θk) = Eθk [log(Lct(θ)) | N ]

•Maximization (M) step: θk+1 = arg maxθQ(θ|θk)

θk −→k→∞

local likelihood maxima

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Lct(θ) =

Nt∏n=1

λ(Atn)

e− ∫ tc λ(As) ds

mt∏m=1

pBm(1− p)1−Bm

where As is the virtual age, ie the time elapsed since the last perfect PM

∀s ∈]τm, τm+1] ∩ [c, T ] λ(As) =

m∏i=0

[λ(s− τm−i)]1l{B−iτm}

where B−iτm =“at time τm, the last AGAN PM time is τm−i”

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Eθ[log(Lct(θ))|N ]=Q(p|θ)︷︸︸︷Lct(θ) =

Nt∏n=1

λ(Atn)


︸︷︷︸Eθ[log( . )|N ]=Q2(λ|θ)

mt∏m=1

pBm(1− p)1−Bm

︸︷︷︸

Eθ[log( . )|N ]=Q1(p|θ)


∀s ∈]τm, τm+1] ∩ [c, T ] λ(As) =

m∏i=0



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Eθ[log(Lct(θ))|N ]=Q(p|θ)︷︸︸︷Lct(θ) =

Nt∏n=1

λ(Atn)


︸︷︷︸Eθ[log( . )|N ]=Q2(λ|θ)

mt∏m=1

pBm(1− p)1−Bm

︸︷︷︸

Eθ[log( . )|N ]=Q1(p|θ)


∀s ∈]τm, τm+1] ∩ [c, T ] λ(As) =

m∏i=0



• Q2 function of πθm,i = Eθ[1l{B−iτm}

| NT = nT , TnT = tnT

]• Q1 function of πθm,0 = πθm = Eθ

[Bm | NT = nT , TnT = tnT

]16 of 28


E step: Compute for 0 ≤ m ≤ mT and 0 ≤ i ≤ m,

πθm,i = Eθ[1l{B−iτm}

| NT = nT , TnT = tnT

]• π0,0(θ) = 1

• for m ∈ {1, ...,mT}, πθm,0 = πθm

• for 0 ≤ m < mT and 0 ≤ i ≤ m,

πθm+1,i+1 = πθm,i − p1+1l{m>i}(1− p)i

∏(c∨τm−i)<tj≤τm+1

λ(tj − τm−i)

e−(Λ((c∨τm+1)−τm−i)−Λ((c∨τm−i)−τm−i))

Lc∨τm−i(θ)Lτm+1,T (θ)

LT (θ)

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M step: λ(t) = αβtβ−1

• pk+1 =[∑mT

m=1 πθkm,0

]/mT

• βk+1 = arg maxβ

[nT (log(nT β)− log(Sk(β))) + (β − 1) ∑

τ∈{τmc+1,...,τmT−,T}

mτ−∑i=0

πθkmτ−,mτ−−i

∑(c∨τm

τ−)<tj≤τ

log(tj − τi)

]

with Sk(β) =∑

τ∈{τmc+1,...,τmT−,T}

mτ−∑i=0

πθkmτ−,mτ−−i

((τ − τi)β − ((c ∨ τmτ−)− τi)β)

• αk+1 = nT/Sk(βk+1)18 of 28


VI. Reliability indicators

• Failure intensity: λt = lim∆t→01

∆tP (Nt+∆t −Nt− |TNt−, Nt−)

λt =

∑mt−m=0D

mCK

t−e−(Λ(t−τm)−Λ((c∨τm)−τm)Lc∨τm(θ)∑mt−

m=0DmCK

t−λ(t− τm) e−(Λ(t−τm)−Λ((c∨τm)−τm)Lc∨τm(θ)

•Cumulative failure intensity: Λt =∫ t

0 λs ds

Λt = −

Kt∑k=1

log

LC−k(θ)

LC+k−1

(θ)

− log Lt−(θ)

LC+Kt

(θ)

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•Reliability: P (TNT+1 > s |TNT , NT ) = exp(−(Λs − ΛT ))

• Expected cumulative number of failures: E[Ns |TNT , NT ] =

NT +

ms∑i=mT+1

(Λ(s− τi) + E[Nτ−i|TNT , NT ]−NT )p(1− p)ms−i

+

[ mT∑i=0

(Λ(s− τi)− Λ(T − τi))(1− p)ms−mTπθmT ,mT−i

]

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VII. Application

The BP PM-ABAO CM model is implemented in MARS (MaintenanceAssessment of Repairable Systems): a free software developped byLJK (Grenoble university) and EDF (French electricity utility).

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•MLE combined with moment estimation:

α = 1.84×10−6, β = 1.95, p = 0.602, log(LT (α, β, p)) = −150.900

•MLE computed with EM algorithm or direct maximization:

α = 1.87×10−6, β = 1.94, p = 0.614, log(LT (α, β, p)) = −150.902

0 2000 4000 6000 8000 10000 12000 140000

5

10

15

20

25

πθm: 0.61 0.62 0.69 0.73 0.07 0.00 1.00 0.99 0.96 0.46ABAO ABAO AGAN AGAN AGAN

Wearing out state at time c ⇒ failures surge

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Quality of estimation: θ = (1.9e− 6, 1.9, 0.61)

NEB: Normalized Empirical BiasNESD: Norm. Emp. Standard Deviation

}⇒ estimated over

60 000 simulationsθ∗: MLE estimator when the Bi are known

Initial intensity: λ(t) = αβtβ−1 or λ(t) = β/η(t/η)β−1

EM is more robust than direct likelihood maximization

NEB NESDα η β p α η β p

θ 590% 36% 8.2% -9.4% 3300% 130% 35% 50%

θ 580% 37% 9.2% -12% 3300% 130% 36% 51%

θ∗ 18000% -24% -12% 0% 5000% 35% 22% 25%

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Quality of individual PM efficiency estimation:

Empirical mean Empirical standard deviation

mth PM πθm −Bm p−Bm πθm −Bm p−Bm1 -7.8 e-2 -5.4 e-2 5.8 e-1 5.8 e-12 -9.3 e-2 -5.8 e-2 5.9 e-1 5.8 e-13 -9.4 e-2 -5.8 e-2 5.9 e-1 5.7 e-14 -6.8 e-2 -5.8 e-2 6.0 e-1 5.6 e-15 -4.3 e-2 -5.6 e-2 5.3 e-1 5.2 e-16 -3.7 e-1 -6.1 e-2 6.4 e-1 5.8 e-17 -1.3 e-2 -5.8 e-2 2.8 e-1 4.6 e-18 7.2 e-3 -5.8 e-2 3.9 e-1 5.1 e-19 9.4 e-4 -5.4 e-2 4.0 e-1 5.2 e-110 -8.5 e-3 -5.5 e-2 4.7 e-1 5.5 e-1

Mean -7.5 e-2 -5.7 e-2 5.1 e-1 5.4 e-1

⇒ Individual PM efficiency estimation is relevent after c24 of 28


Failure intensity for θ = θ

0.9 1 1.1 1.2 1.3 1.40

0.002

0.004

0.006

0.008

0.01

Cumulative failure intensity and number of failures for θ = θ

0.9 1 1.1 1.2 1.3 1.40

5

10

15

20

25

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Forecast reliability for θ = θ

1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95

x 104

0

0.2

0.4

0.6

0.8

1

Forecast cumulative number of failures for θ = θ

1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95

x 104

24

28

32

36

40

44

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VIII. Prospects

•Maintenance times optimization

• Consider a more general distribution over ]−∞, 1] for the Bi.

• Develop confidence intervals and tests for the BP model

• ...

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References

• Agustin MZN and Pena EA. Order statistics properties, random generation, and goodness-of-fit testing for a minimalrepair model. Journal of American Statistical Association 1999; 94: 266-272.

• Bathe F and Franz J. Modeling of repairable systems with various degrees of repair. Metrika 1996; 43: 149-164.

• Brown M and Proschan F. Imperfect repair. Journal of Applied Probability 1983; 20: 851-859.

• Kumar U and Klefsjo B. Reliability analysis of hydraulic systems of LHD machines using the power law processmodel. Reliability Engineering and System Safety 1992; 35: 217-224.

• Kvam PH, Singh H and Whitaker LR. Estimating distributions with increasing failure rate in an imperfect repairmodel. Lifetime Data Analysis 2002; 8: 53-6.

• Hollander M, Presnell P and Sethuraman J. Nonparametric methods for imperfect repair models. The Annals ofStatistics 1992; 20: 879-896.

• Langseth H and Lindqvist BH. A maintenance model for components exposed to several failure mechanisms andimperfect repair. In Mathematical and Statistical Methods in Reliability, B.H. Lindqvist, K.L. Doksum (eds). WorldScientific: 2004; 415-430.

• Lim JH, Lu KL, Park DH. Bayesian imperfect repair model. Communications in Statistics - Theory and Methods1998; 27: 965-984.

• Lim TJ. Estimating system reliability with fully masked data under Brown-Proschan imperfect repair model. Reli-ability Engineering and System Safety 1998; 59: 277-289.

• Lim TJ and Lie CH. Analysis of system reliability with dependent repair modes. IEEE Transactions on reliability2000; 49: 153-162.

• Whitaker LR and Samaniego FJ. Estimating the reliability of systems subject to imperfect repair. Journal ofAmerican Statistical Association 1989; 84: 301-309.

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A generalized Brown-Proschan model for preventive and [email protected] Jean Kuntzmann Laboratory Grenoble University France 1 of 28. ISBIS-2010 Laurent DOYEN