Abstract. This paper addresses the problem of generating optimal inspection strategies for randomly failing systems whose state is known only through ...
INT. J. CO MPUTER INTEGRATED MANUFACTURING, 1 998, VO L. 11, NO . 3, 226 ± 23 1
In spe ction an d pre dictive m ain te n an ce strate gie s ANIS CH ELBI an d DAO UD AIT-KADI
Ab stract. Th is pape r addre sse s the proble m of gene rating op tim al inspe ction strategie s for random ly failing syste m s whose state is kn own on ly throu gh inspection. Two inspe ction strategies are presente d. Th e first on e is a sim ple inspe ction strategy while the secon d is a pre dictive orie nted on e. Both strategies are asse ssed on the basis of the ave rage total cost pe r tim e un it ove r an infinite span . For a give n e qu ipm e nt lifetim e distribution fun ction, a n um e rical exam ple is pe rform e d and re com m endations con ce rning the choice be twee n the two strategies are form ulate d.
1. In tro d u c tio n A large n um be r of m ac hin e com pone n ts such as ge ars, be arin gs, valve s, e tc. are subje ct to ran dom failu re . In m an y case s, th e state of such ran dom ly failin g com pon e nts is known on ly through inspe ction ( Barlow an d Prosch an 1965) . Afte r e ach in spe ction , th e e quipm e nt is e ith e r re place d if foun d in a faile d state, or le ft in ope ration. In orde r to ke e p e quip m e nt in ope ration , on e has to build an in spe ction strate gy wh ich spe cifies the te stin g se que nce . Th is strate gy m ust conside r th e com pone n t’ s life tim e distribution , the cost in curre d to pe rform an inspe ction an d th e cost re late d to th e in activity pe riod be twe e n failure occurre nce an d its de tection . Barlow an d Proschan ( 19 65) de rive d an optim al inspe ction strate gy de fin e d as an in spe ction se qu e n ce wh ich m in im ize s the total e xpe c ted cost pe r tim e unit ove r an in fin ite span . Th e proc e dure th e y de ve lop e d turn e d out to be difficult to im ple m e nt. Th e com putational difficultie s arise from using the inspe ction in stan ts {x1 ,x2 , ...,xn } as th e con trol variab le s. Nakagawa an d Yasu i ( 1980) de ve lope d a n e arly optim al inspe ction strate gy re duc in g the com putation al difficultie s by com p utin g succe ssive in spe ction in stan ts backwards
Au thors: A. Ch e lbi an d D. Ait-Kadi, De partm e n t of Me chan ical Engin e e rin g, Laval Un iversity, Q u e be c, P.Q , G1K 7P4, Can ad a. E-m ail:ach e lbi@gmc .u laval. ca; daoud a@gm c.ulaval.ca 0951-192X / 98 $12 × 00
starting from a large inspe ction in stant xn . Particular case s of pe riodic inspe ction s h ave be e n ad dre sse d by m an y auth ors ( Rodrigu e s 1990, Schn e e we iss 1976 an d Tad ikam alla 1979 ) . In th is pap e r, two in sp e ction strate gie s are propose d. Th e first on e , pre se nte d in Section 2 sugge sts in spe cting the e quipm e n t at pre de te rm in e d instan ts. If failu re is de te cte d the faile d com pon e n t is im m e diate ly r e p lac e d . Th e se c on d strate gy ( in Se c tion 3 ) is pre dictive orie nte d, an d consists in fixin g on e or m an y con trol param e ters wh ich can de fin e th e actual state of th e e quip m e nt com pon e n ts, an d startin g pre ve n tive action s on th e basis of an an alysis of pre de te rm in e d con trol param e te r te n de n cie s. For e ach strate gy, th e re late d assum ption s as we ll as th e m ath e m atical m ode l are pre se n ted. Th e pe rform an ce of e ach strate gy is e valuate d ove r th e e quipm e n t life cycle . In both case s, th e seque n ce of inspe ction tim e s is th e on e wh ich m in im ize s the total ave rag e cost pe r tim e unit ove r an in fin ite span . Th is cost re pre sen ts th e pe rform an ce crite rion sele cte d to ch oose th e be st suite d strate gy for a give n e quip m e nt un de r re le van t costs. In Se c tion 4 , we sh ow, th rou gh a n um e rical e xam ple , h ow th e ch oice be twe e n th e two strate gie s c an be m ad e in th e c ase of a ran d om ly failin g e quipm e n t wh ose life tim e is n orm ally distribute d.
2. Sim p le in sp e c tio n strate gy Th is strate gy sugge sts th at the e quipm e n t m ust be in spe cted accordin g to a pre de te rm ine d inspe ction se qu e n c e xi ( i 5 1 2 . . .) to de te rm in e if it is still ope rating or pre sen ts som e form of failu re . Th e m ore fre que n tly the e quipm e n t is in spe cted th e shorte r th e pe riod be twe e n failure an d its de tec tion. O n the othe r h an d, e ach in spe ction involve s a cost so that we do n ot wish to in spe ct too ofte n . Th e proble m is to de te rm in e th e se que n ce of in spe ction in stants wh ich m in im ize s th e ave rage cost pe r tim e unit. 1998 Taylor & F rancis Ltd
In spection an d predictiv e main ten an ce strategies Conside ring x0 5
Th e followin g assum ption s are m ad e : ( a) ( b) ( c) ( d) ( e)
syste m failu re is known on ly th rough in spe ction; inspe ction take s n e gligible tim e ; the in spe ction action is pe rfectly pe rform e d; e ach in spe ction im plie s a fixe d cost Ci ; a cost Cd is incurre d to e ach tim e un it e lap se d be twe e n failu re occurre n ce an d its de te ction; ( f) upon de tec tion of failure , re pair ( or re place m e nt) re quiring ne gligible tim e is un de rtake n at an ave rage cost Cg ; ( g) the system life tim e s is a ran d om var iable with kn own probability de n sity function g( .) an d distribution fun ction G( .) ; ( h) th e system failu re rate is an incre asin g function.
Th e propose d m ath e m atical m ode l for th is strate gy is an e xten sion of th e one de ve lope d by Mun ford an d Sh ah an i ( 1972) wh ic h p ropose s a n e arly optim al inspe ction policy base d on a single param e te r p. Th is param e te r stan ds for th e con dition al probability of failu re occurre n ce durin g the in terval [ 2 1 ] give n th at th e e quipm e n t was workin g at tim e 2 1: G ( xi ) 2 G ( xi 2 1 ) 1 2 G ( xi 2 1 )
for i 5
1 2
...
pi 5
p1
i
5
G ( xi ) 2 G ( xi 2 1 ) 1 2 G ( xi 2 1 )
for i 5
1 2
...
( 2)
wh e re p 1 is th e condition al probability that failu re occu rs be fore th e first in sp e ction . Th is form was arbitrarily ch ose n . O the r form s wh ich satisfy th e con dition: pi 1 m ay be use d.
1
pi
p1 5
0, (4)
G ( x1 )
from e quation ( 2) , we ge t G ( xi ) 5
pi 1
(1
2 p i ) G ( xi 2 1 )
for i 5
1 2
...
(5)
an d the inspe ction con stan ts xi can e asily be ge n e rate d with the followin g re cursive form ula: xi 5
G2
1
1
p1
12
i
G ( xi 2 for i 5
1 G ( xi 2 1 )
1)
...
1 2
(6)
Le t TC be th e total cost un til failu re is de te cte d an d re place m e n t is m ade ( cycle cost) , iCi 1
TC 5
for i 5
1 2
...
R 5
Cd ( xi 2
( 3)
E ( TC )
lim
®
)
1 Cg
(7)
¥
E( C )
5
(8)
E( T )
wh e re E( C) is the e xp e cted total c ost ove r a re place m e nt cycle wh ose ave rage duration is E ( T) . E( C) an d E( T) are , re spe ctive ly, give n by Ci E ( I ) 1
E( C) 5
( 1)
Th e inspe ction in stan ts ( 5 1 2 . . .) are de rive d as a fun ction of th e param e ter p wh ich is ch ose n so as to m in im ize th e ave rage cost ove r a cycle ( th e e lap se d tim e be twe e n con se c utive re place m e n ts) . Con sid e rin g th at the e quipm e n t failure rate is an incre asin g function ( assum ption ( h ) ) , wh ich m e an s th at failu re re pre se n ts th e outcom e of a de terioration proce ss th at grows with tim e , we se e that as tim e e lap se s an d in spe ction s are carried out, the probability of failu re occurre nc e incre ase s. In oth e r words p grows toge th e r with i. th e con dition al probability p i th at failu re occurs during th e inte rval [ xi 2 1 xi ] give n th at th e e quipm e n t was in ope ration at tim e xi 2 1, m ay th e n be e xp re sse d as follows: 1
0 an d G ( 0 ) 5
wh e re i is the total n um be r of in spe ction s during on e cycle . Th e e xp re ssion of th e ave rage total cost pe r tim e un it ove r an in fin ite span is give n by:
2.1. Mathematical model
p 5
227
Cd E ( A) 1
(9)
Cg
1 E ( A)
E( T ) 5
( 10 )
wh e re E( I) is the ave rage n um be r of in spe ction s un til failu re de te c tion, E( A) the m e an inactivity pe riod, an d l th e e quipm e n t ave rage life tim e . Le t us de ve lop e xp re ssion for E( I) an d E( A) .
The av erage n umber of in spection s du ring on e cycle, E(I) If th e ran d om var iable I de n ote s th e nu m be r of in spe ctions until failu re de te c tion, th e n th e e xp e cte d n um be r of in spe ction s is give n by:
¥
E( I ) 5
i Prob( I 5
( 11 )
i)
i5 1
wh e re p1
Prob( I 5
i2 1
i) 5
1 k
.p11
1 2 p1
k5 1
i
for i 5 for i 5
1 2 3
... ( 12 )
He n ce , E( I ) 5
p1 1
¥ i5 2
1 i
i . p1
.
i2 1 k5 1
12
1 k
p1
( 13 )
A. Chelbi an d D. Ait-Kadi
228
The av erage in activ ity period, E(A) If failu re occurs be twe e n t an d t+dt, with in th e tim e inte rval [ xi-1 , xi] , th e n th e m e an inactivity pe riod is give n by: xi
A5
2 t ) g ( t ) dt
( xi
( 14 )
xi 2 1
Sin ce failu re m ay occur with in an y inte rval [ xi-1 , xi] , th e n
¥
E ( A) 5
xi
( xi i5 1
2 t ) g ( t ) dt
( 15 )
xi 2 1
wh e r e x0 =0. Afte r som e alg e braic m an ipu lation s, e quation ( 15) m ay be put in th e followin g form :
¥
E ( A) 5
i5 1
xi [ G ( xi ) 2
From e quation ( 2) , we ge t G ( xi ) 2
G ( xi 2
th e re fore E ( A) 5
1)
¥
1 i
5
p1
i2 1
1 i
i5 1
xi p 1
k5 1
i2 1
12
k5 1
12
2
1)]
G ( xi 2
( 16 )
1 k
p1
1 k
2
p1
( 17 )
( 18 )
s : tim e at wh ich the m e asure d param e ter e xce e ds
th us, substitutin g in e quation s ( 8) , ( 9) an d ( 10) , th e e xp re ssion for th e ave rage total cost pe r tim e un it ove r an in fin ite span be com e s: Ci
¥
p1 1
1 i
i5 2
i p1
i2 1
1 k
1 i
xi p 1 i
R ( p1 ) 5
1 i
i
xi p 1
1 Cd
1 2 p1
k5 1
i2 1 k5 1
i2 1 k5 1
an an alysis of the tre nd of pre d e te rm in e d con trol param e ters wh ich can de fin e th e actual condition of th e e quipm e nt com pon e n ts. Vibration le ve l an d th e am oun t of carbon in a lubrican t are two e xam ple s of such con trol param e te rs. Th e m ath e m atical m od e l assoc iate d with th is strategy h as be e n discussed in Ch e lbi an d Ait-Kad i ( 1995 ) an d Tu rco an d Parolin i ( 198 4) . It ge n e rate s an in spe ction se que n ce m in im izin g th e total ave rag e cost pe r tim e unit, wh ich is th e sum of costs re late d to in spe c tion s, pr odu c tion losse s an d p re ve n tive or corre ctive action s. The m ode l states th at progre ssion of th e e quip m e n t de te rioratio n p r oce ss c h an ge s sign ifican tly on ce a thre sh old pe rform an ce ( sh own by m e asu re d value s of the param e te rs unde r con trol) is re ach e d. Afte r this critical instan t, the de te rioration rate in cre ase s m ore quickly. For th is re ason , it is assum e d that th e e quipm e n t failu re rate ch an ge s on ce th e alarm th re sh old is e xce e de d. In th is situation , failu re be com e s im m ine n t. The probability de nsity fun c tion associate d to th e e quipm e n t life tim e is th e re fore con side re d as a com bination of two probability de n sity fun ctions / ( .) an d f( .) , re spe ctive ly, associate d to two in de pe n de nt ran dom variab le s s an d t:
1 k
1 2 p1
t:
The followin g costs are con side re d: 2
1 Cg
( 19 )
12
th e alar m th re shold, from th e in stan t wh e n th e e quipm e n t is re store d failu re tim e from in stan t s up to failure
1 k
p1
Th is e xpre ssion is re lative ly com ple x. It in clude s th e p ar am e te r p 1 as we ll as th e in sp e c tio n in stan ts xi ( i 5 1 2 . . .) . For a give n value of p 1 , it is possible to com pute th e corre spon din g instan ts xi from e quation ( 6) . A n um e rical ite rative proce dure h as be e n de ve lope d to ge ne rate th e optim al value of p 1 corre spon din g to th e se que n ce xi ( i 5 1 2 . . .) wh ich m in im ize s th e total e xp e cted cost pe r tim e unit, R,( p 1 ) . Th e user h as sim ply to input the e quipm e nt life tim e distribution G( .) as we ll as th e re le van t costs Ci , Cd an d Cg , an d a n e arly optim al in spe ction se que n ce is obtain e d.
Cv Cp Cg Cd
in spe ction cost, pre ve ntive re place m e n t cost, failu re re place m e n t cost, cost of an inactivity tim e un it
The m ode l m ake s th e followin g assum ption s ( 1) ( 2)
( 3)
3. Pre d ictive -o rie n te d in sp e ction strate gy Th is strate gy is base d on th e syste m atic con trol of th e e quipm e n t. Pre ve n tive action starts on the basis of
: : : :
( 4) ( 5)
T h e p r o b ab i l i t y d i s tr i b u ti o n fu n c ti o n s ( ) an d ( ) are kn own . In spe c tion s h ave n o c on se que n ce on th e physical proce ss of de te rioration an d failure ( pe rfe ct in spe ction ) . The possible pre ve n tive re place m e n t is sch e dule d afte r a pre de te rm in e d tim e H from th e in spe ction in stan t in wh ic h th e param e te r unde r con trol is foun d havin g e xce e de d th e alarm le ve l. An y possible in spe ction with in th is tim e H is can ce lle d. In spe ction s do n ot ne e d an y se tup tim e . The e quipm e n t con dition in cludin g failu re is kn own only through inspe ction .
In spection an d predictiv e main ten an ce strategies Th e proble m is to fin d the instants xi ( i 5 1 2 . . .) wh ich can give th e lowe st ave rag e cost pe r tim e un it, th at is the ratio be twe e n cycle ave rage cost E( C) an d cycle ave rage tim e E( T) ove r an in fin ite h orizon. He re , a cycle re pre se nts the e lap se d tim e be twe e n con se cutive re place m e n ts, e ithe r pre ve ntive or due to failu re . Th e m ode l conside rs th e sam e e xp re ssion for p i ( e quation ( 2) ) to e xp re ss th e con dition al probability to th de te ct at th e i in spe ction , th at th e alarm thre sh old h as be e n e xc e e d e d give n th at th is situatio n was n ot th discove re d at the ( i-1) in spe ction . Th e inspe ction se que nce can be de term ine d for a give n p 1 by: xi 5
2 1
1 i
p 1 [1 2
1)] 1
( xi 2
( xi 2
1 2
...
( 20 )
p1 be in g in th is case , th e probability of havin g th e alarm th re sh old e xce e de d be fore the first in spe ction. Th e ap propriate p 1 will be the on e wh ich m in im ize s th e ratio E [C ( p 1 ) ] ( 21 ) 5 E [T ( p 1 ) ] wh e re E[ C( p1 ) ] is th e cycle ave rag e cost an d E [ T ( p 1) ] is th e cycle ave rage tim e . Th e e xpre ssion for th e cycle ave rag e cost is: Cv E [ I ( p 1 ) ] 1
E [C ( p 1 ) ] 5
Cg Pg ( p1 ) 1
Cp Pp ( p 1 ) 1
( 22 )
Cd A( p 1 )
wh e re th e probability, Pg, that the cycle e nds with a failu re e ve nt is give n by
¥
Pg 5
xi
( )F(H i5 1
xi 5
1 xi 2
)d
( 23 )
1
Th e probability, Pp , th at the cycle e n ds with a pre ve n tive inte rve n tion is give n by Pp 5
12
( ) ( xi
1 H 2 ( 1 t ) ) f ( t ) dt }d
( 26 )
O n th e oth e r h an d, th e e xp re ssion for the cycle ave rage tim e E[ T( p 1 ) ] is d e rive d from th e two followin g obse rvations: If failu re occurs be twe e n xi an d xi+H, the cycle ave rage duration would be give n by
¥
E1 [ T ( p 1 ) ] 5
xi 1 H 2
xi xi 2 1
( )
i5 1
xi 1
0
H
f ( t ) dt d
( 27 )
If failu re doe s n ot occur with in the tim e in te rval [ xi, xi + H] , in wh ich case a pre ve n tive re place m e nt is m ad e , th e cycle ave rage duration would be give n by
1)
for i 5
{
229
( 24 )
Pg
E2 [ T ( p 1 ) ] 5
¥ (H i5 1
[1 2
1 xi )
F (H 1
xi xi 2 1
xi 2
( ) ) ]d
( 28 )
Sin c e th e ab ove m e n tion e d e ve n ts are m utu ally e xclusive , th e cycle e xpe cte d duration is th e n: E [ T ( p1 ) ] 5
E1 [ T ( p 1 ) ] 1
E2 [ T ( p 1 ) ]
( 29 )
He n ce , com bin in g e quation s ( 21) to ( 29) , it is possible to e xp re ss th e ave rage total cost pe r tim e un it , as a fun ction of a single param e ter p 1 , re placin g in e ach e quation xi by its valu e give n by e quation ( 20 ) . Due to the com ple xity of th e m ode l, a num e rical ite rative proce dure h as be e n de ve lope d to ge n e rate a n e arly optim al inspe ction seque n ce xi ( i 5 1 2 . . .) wh ich corre sponds to the value of p 1 wh ich m inim ize s th e ave rage total cost pe r tim e un it. The in put of th e proce dure include s the probability distribution s F an d F, the costs Ci , Cd , Cg an d Cp , an d the duration H. A se nsitivity an alysis m ay be pe rform e d for e ach strategy usin g th e corre spon din g n um e rical proce dure in orde r to study the in flue n ce of e ach input param e te r on th e inspe ction se que n ce an d its cost.
Th e ave rage num be r of in spe ction s un til the first alarm signal, E[ I( p 1 ) ] , is e xp re sse d as follows
¥
E [ I ( p1 ) ] 5
xi 1
i[
1
( ) . F ( xi 1
o
i5 1 xi
0
( ) F ( xi
2
1
2
)d
)d ]
an d A( p 1 ) is th e ave rage in activity pe riod A( p 1 ) 5
¥
i5 1
xi 1 H 2
xi xi 2
1
0
4. Nu m e rical e xam p le
2 ( 25)
As an application of the two in spe ction strate gie s pre se n ted in th is pap e r, le t us conside r an e quipm e n t wh ose life tim e is a ran dom var iable with norm al distribution G $ ( l =900 , r =90) . Th is e quipm e nt is subm itted to a sim ple in spe ction strate gy in volvin g th e followin g costs: Ci =1 35 unit cost ( cost of one in spe ction)
A. Chelbi an d D. Ait-Kadi
230
Cd =20 un it cost/ un it tim e ( cost of an inactivity tim e unit) Th e se life tim e distribution an d costs data have be e n arbitrarily chose n in orde r to che ck th e valid ity an d th e robustn e ss of th e proc e dure s. Ac c ord in g to th e c orre sp on d in g m ath e m atic al m ode l ( Se ction 2) , th e optim al in spe ction se que n ce wh ich m in im ize s th e total ave rag e cost pe r tim e un it is sh own in Table 1. Th e tim e to pe rform the first inspe ction ( x1 =81 4 × 12 tim e un its) com e s a little be fore th e e quipm e nt ave rage life tim e is re ach e d. Th e n , as we should e xp e c t, since th e e quipm e nt failu re rate is an in cre asin g function , th e inte rval be twe e n succe ssive inspe ction s de cre ase s. Th e corre spondin g total ave rage cost pe r tim e un it is e qual to 7 × 345 ( un it cost/ un it tim e ) . Th e probability p 1 th at failu re occurs be fore the first in spe ction is 17 % . Suppose that on e con trol param e te r, vibration le ve l for instan ce , turn s out to be a good in dication of th e e quipm e nt actual state an d th at it can be m e asure d by m e an s of spe cialize d e quipm e nt. It is th e n worth se e in g if the diagn ostic pre d ictive strate gy ( Sec tion 3) prove s to be a good alte rn ative . In te rm s of costs, th e diffe re n ce be twe e n th e two strate gie s re side s obviously in th e cost of one in spe ction. In de e d, ad optin g th e se cond strate gy would involve an incre ase in the inspe ction cost be cau se of th e n e e d for m ore skille d te chn ical staff as we ll as high tech n ology e qu ip m e n t. Le t us supp ose in ou r case th at th e inspe ction cost wh e n ad optin g this strate gy is twice th e cost in th e first strate gy : Cv =270 unit cost. Assum e th at a ce rtain alarm th re sh old is fixe d an d th at th e e quipm e nt tim e to failu re turn s out to be a sum of two oth e r n orm al distributions F $ ( l 1 =800 , r 1 =78 ) unit tim e an d F $( l 2 =100, r 2 =45 ) un it tim e , be fore an d afte r e xce e din g th e alarm th re sh old. Th e calculation s
corre spon ding to the se con d strate gy ( with H =O , Cp =3500 an d Cg =60 00) yie ld th e followin g re sults. Th e ave rag e total cost pe r tim e unit is e qual to 5 × 396 ( un it cost/ un it tim e ) wh ich is 27 % lowe r th an with th e first strate gy. The probability to have th e alarm th re sh old e xce e de d be fore the first in spe ction is e qual to 40% an d th e optim al in spe ction se que n ce is shown in Table 2. He n ce , on the basis of th e pe rform an ce crite rion we h ave fixe d, we can con clude th at in this case , e ve n if th e cost of an in spe ction double s wh e n switching to a pre dictive in spe ction strate gy, it is still worth doin g it. O n e can ask the followin g que stion: up to wh at am oun t is it still worth in ve stin g? In orde r to an swe r th is que stion , th e grap h ic pre se n ted in Figu re 1 sh ows the variation of the lowe st ave rag e cost pe r tim e un it c orre spon din g to th e pre dictive in spe ction strategy, as a fun ction of th e ratio R =Cv / Ci/ ( Ci is ke pt con stan t ( 13 5 unit cost) an d Cv is varie d) . O n e can se e from th is figure that as lon g as th e
Table 2. O ptim al inspe ction seque nce correspon ding to th e se con d strategy. i 0 1 2 3 4 5 6 7 8 9 10
x
i
0 780 × 35 860 × 19 922 × 70 976 × 94 1027 × 64 1072 × 76 1115 × 20 1151 × 05 1181 × 36 1200 × 21
x
i +1
Ð
x
i
780 × 35 79 × 84 62 × 51 54 × 24 50 × 70 45 × 12 42 × 44 35 × 85 30 × 31 18 × 85
Table 1. O ptim al inspection se que nce corre spon ding to the first strategy. i 0 1 2 3 4 5 6 7 8 9 10
xi 0 814 × 12 880 × 00 912 × 24 932 × 79 947 × 64 959 × 13 968 × 44 976 × 22 982 × 88 988 × 68
xi +1 Ð
xi
814 × 12 65 × 88 32 × 24 20 × 55 14 × 85 11 × 49 9 × 31 7 × 78 6 × 66 5 × 80
Figure 1. Variation of the lowest average cost corresponding to the predictive strategy function of the inspection costs ratio.
In spection an d predictiv e main ten an ce strategies inspe ction cost in the case of a pre dictive strate gy is le ss th an twe lve tim e s the cost of a sim ple in spe ction , base d on our pe rform an ce crite rion , it is re com m e n de d to ad opt th e pre dictive in spe ction strate gy.
5.
Co nclu sio n
In this pap e r, two in spe ction strate gie s h ave be e n pre se n te d. Th e first on e can be ad van tage ously ap plie d to an yt e quipm e n t subje ct to ran dom failu re s an d wh ose state is kn own only th rough inspe ction . Th e se con d strate gy in troduce s th e pre dic tive m ain te nan ce con ce pt. In fact, at e ach inspe ction , th e ten de nc y of a m e asu rable control param e te r of the syste m is an alyse d an d it is d e c id e d wh e th e r to m ake a p re ve n tive re place m e nt or not. Give n the e quipm e nt life tim e probability distributions as we ll as th e costs associate d to the diffe re nt action s, th e m ode ls corre spon din g to the two propose d strate gie s ge n e rate a n e arly optim al in sp e ction se que n ce . Th e n um e rical re sults obtain e d are in accordan ce with wh at h as be e n alre ad y publishe d on the subje ct. For give n value s of e ach m ode l’ s param e te rs, th e user can ch oose the strate gy with th e lowe st total ave rage cost pe r tim e un it on an in fin ite span . Q uality con trol an d m e dical te sts re pre se nt som e pote n tial practical application s of th e se m ode ls.
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Ackn o wle d ge m e n ts Th is re se arc h is partially su pporte d by th e Natural Scie nce an d En gin e e rin g Re se arch Counc il of Can cad a ( Grant O GP 0118 062) j. The au thors are ple ase d to th an k th e asscociate e ditor, F. Ve rnad at, an d th e re fe re e s for th e ir insigh tful com m e n ts.
References B ARLO W, R. E., and PRO SCH AN, F., 1965, Mathematical Theory of Reliability ( Joh n Wiley & Son s, Ne w York) . C H ELBI , A., an d AIT-KADI, D., 1995, O ptim al con dition al re placem e nt m ode l. Te chn ical Re port, De partm en t of Me chanical En gine e ring, Laval Un ive rsity, Q ue be c, Canada. MUNFO RD, A., and SHAH ANI, A., 1972, A ne arly op tim al inspe ction policy. Operational Research Qu arterly, 23, 373 ± 379. NAKAGAWA, I., and YASUI., K., 1980, Approxim ate calculation of op tim al inspection P83FD. Jou rn al of Operational Research Society, 31, 851 ± 853. RO DRIGUES DIAS, J., 1990, A new approxim ation for inspe ction pe riod of system s with differe nt failure rate s. Eu ropean Jou rn al of Operational Research, 45, 219 ± 223. S CH NEEWEISS , W. G., 1976, O n the m e an du ration of hidden faults in pe riodically che cke d system s. IEEE Tran saction s of Reliability, R-25 / 5, 346 ± 348. T ADIKAMALLA, P. R., 1979, An inspe ction policy for the Gam m a Failure Distribution s. Jou rn al of Operational Research Society, 30, 77 ± 80. T URCO , T., an d PARO LINI, P., 1984, A ne arly optim al inspe ction policy for produ ction e quipm e nt. In ternational Journal of Produ ction Research, 22, 515528.
