Optimal degradation processes control by two-level policies

RELIABILITY ENGINEERING &

SYSTE}I ELSEvIER

ReЦability Engineering and System Safety бб (1999) 1-11

SАFЕтY

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sечiет.соmЛосаtе/rеss

Optimal degradation processes control Ьу two-level policies V.А. Kopnov* РО Вох 64, Ekлterinburg 620107, Russia Received 18 July 1998; accepted б January 1999

Abstract Trvo-level control policies аrе applied to various types ofMarkov рrосеssеs describing degTading рагаmеtеrs of system units. То estimate the system ореrаtiоп cost an expectation of losses реr unit time is evaluated. It is assumed that the degradation process is observable, and а monitoring system can signal about future failures. Fiгstly, semi-Markov рrосеssеs аrе consideгed. А death process is proposed fог а unit subjected to соrrоsiоп. А Маrkоч chain аrе used fоr the problem of fatigue crack gTowth. The control рrоЬlеm is sfudied mainly in а steadystate regime fоr units of multiple use when the cost function is реrfоrmеd as the ratio of expectations. The cost function as the expectation of losses реr unit time fоr single units is also studied, and the majorizing рrореrtу of this cost function is shown. О 1999 Elsevier Science Ltd.

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rights rеsеrчеd.

Keywords: Two-level policy; Optimal conffol; Мшkоч processes; Degradation pfocesses

when stopping and replacement are implemerrted. The rесочеry of such а unit is also possible-the crack mау Ье

1. Introduction

The well-known (s,S) policies in the inventory problem [1] can Ье used for maintenance in mechanical engineering, though some adaptations shоцld Ье mаdе. The backgTound of this article was published in Ref. [2] where the tеrm twolevel policies was adopted. The main difference between the two-level control policies appНed to mechanical engineering systems and inventory ones is that fictitious levels when modeling can Ье omitted. It is assumed that а mechanical system contains а dеgrаd-

ing unit. The degradation process is observable, otherwise there is а раrаmеtеf associated with the рrосеss, which can Ье traced, for example, а signal of acoustic emission. The control principles of these policies mау Ье easily explained as follows. Let а unit subjected to corrosion have а рrоtесtive covering, which dесrеаsеs during operation, and it is possible to trасе its thickness. The рrоЬlеm is to find the optimal value of the initial thickness and the thickness when the covering should Ье renewed. If the preventive гепеwаl is late, the failure occurs. Let us give mоrе examples. Let а unit fail due to fatigue сгасk growth. А fatigue gage as а plane notched specimen can Ье fitted, which works with the unit and reflects its damage accumulation process. The problem is to find the optimal чаluе of the gage's сrасk

* Те|.: *7 -З4З2-531'794; fe;r.: +7 -З4З2-24оЗЗ'7

Е-mаil address: [email protected] (V.А. Kopnov) 0951-8з20/99l$ - see front matteT

РII: S095 1-8320(99)00006-х

о

.

welded. It is known фаt фе rпеаr of the tool of mасhiпеtools is correlated rvith thermo-emf of the pair "cutterblank". А monitoring system may Ье installed, and the optimal чаlче of the measured voltage Ье determined to рrечепt failures and оuфuts of рооr quality. The control рrоЬlеm of wеаr of а plain bearing is discussed in Ref. [3]. If the degTadation рrосеss is associated with extemal loads, and а unit fails when the load exceeds а failure level, then partial оr full discharge of loads is another ехаmрlе of sчсh сопtrоl. If degradation is associated with shocks, then there can Ье found some сопlmоп characters with the мо-раrаmеtеr policy studied Ьу Мurрhу and Iskandar [5]. Тhцs, there аге one оr two levels of the degradation process to Ье found оut fгоm an appropriate optimal рrоЬlеm. The first is for the level signalizing about future failure, the second is for failure. It should Ье pointed out that the highest (failure) 1ечеl of degradation is also optimized, but it is not obligatory. In some cases it is inaccessible for control, because it can Ье predetermined Ьу the design standaTds оr оthеr limitationý. The methods described in the following impose also constraints on accessibility of this

level for control modeling. The failure level is closely connected with the reliability of а unit, and this relation

сап Ье very complex. The direct increasing оr decreasing of the level as studied in the article is not always applicable. Modeling of the degradation Ьу Маrkоч рrосеssеs, when

1999 Elsevier Science Ltd. Д1 rights rеsоrчеd.

V.Д. Корпоч / Reliability Епgiпееriпg апd System Sаfеф 66 (1999)

1-1I

fаilчrе

signal

signal

Fig. 1. Possible realization of а controlled semi-Markov process (arrows down denote moments of feeding а signal).

elements of MaTkov matrices do not depend on the failure level, is а certain restriction. Although the control policy is called two-level, there is one mоrе parameter for optimization-the recovery value. The recovery of а unit mау Ье partial оr complete, depending on an engineering рrоЬlеm. Fоr the caseý of corтosion апd load processes, the partial recovery is quite apparent алd We need not make changes in processes descriptions to Stay

in the frame of the proposed methods. In other cases, for example, of fatigue crack gTowth, the degradation behavior of а unit with а welded crack will Ье changed and а new description will Ье required. These considerations show that the rапgе of applications of the two-level control policy is

wider than it is studied in the following. The essence of such а policy is "balancing within sоmе degradation levels" to achieve the optimal reliability of а unit. То derive optimal values of the two-level policy we minimize the expected losses реr unit time. The control рrоЬlеm

is studied mainly for units of multiple use when the cost function is реrfоrmеd as the ratio of expectations of losses and cycle duration in а stationary case. The control рrоЬlеm of units of single usе is also discussed in Section З, and the

difference between the two types

of cost

functions is outlined. It will Ье shown that the cost function of single

unit operation majorizes the cost function of units of multiple use. ThTee sections of the article hаче tfuee levels of models difficulty: fiTst is theoretical, second is mоrе intelligible, and the third model is very sirnple.

constructed firstly оп the base of semi-Markov ргосеssеs with discrete states and continuous time. 2.

]. Semi-Markov process descriptioп

In соmmоп case, the state of degrading units сап Ье described Ьу а semi-Mffkov process Х(/): {хп,{п,п2 0,r > 0}. The first component.Tn is а state of the process after п jumps, which fоrms ап embedded Markov chain and

takes values in the space of nonnegative integer numbers |0,|,2,...}. 6" is а nonnegative сопtiпuоus random чаriаЬlе setting а time interval between Markov rепеwа] moments. The discrete semi-Markov рrосеss is defined Ьу the sеmi-маrkоy mаtriх

У:

QilG): Plxn+l: j,€п+t < t|x": i|,

which can Ье represented as Q,i@ : р;lFц(t) where р;; : Qi1@) : Plxn+t : jlxn : j} аrе transition probabilitiei of the embedded Markov chain, and F,iG): P{€"*t 1 t|xn: i,xn+t

: jl

аrе distribution functions of the response times

of the process X(r) being in states л, -"-j

^п+l

: i

provided that

- J.

2.2. Two-\evel pol icies Assume that the degradation process starts frоm state 0, signal level, and state п is а sйte of failure. А sequence of transitions frоm one state to another can Ье shown gTaphically as in Fig. 1 whеrе а possible realization of the control process is given. Denote the sets state m corresponds to а

2. Sеmi-Маrkоy рrосеssеs

The models of optimal degradation process control is

i,j С Х

W

:

{п,...}, WV

:

V |) W

V.Д. Корпоч / Reliability Епgiпееriпg оп,d System Sаfеф 66 (1999) 1-11

When the procesý enters set У, а signal is given for preventive rесочеry. А lag time т, Ет 1оо, for rесочеry of the unit is introduced. The rесочеry itself оссurs instantly of чаluе А, А < n. When the process enters set IV', the unit fails, and the system fails possibly too. After а delay т1, Ет1{ оо, the unit is rерlасеd. The following notations for transition times are adopted:

Zi,yy

:С f{t,X(t) с

Zу,у:€f

{t,x(t)

:с

f{t,x{t) €

Z;,y

с

УI4llх(0)

l4llx(0) I4l|x(0)

€

: i, i с

U},

у},

: i, i €

i.e.

P{Zy,y:

0} >

(1)

UV|.

0.

(2)

colTesponding to the signal level, ll colTesponding to the failure level and rесочеry value Д. ,1l

process X"(r) as shown in Fig. 1. The moments of recovery of the unit аrе considered as moments of regeneration, and periods of regenerative cycles |0u i: |,2,...| fоrm а rесurrent rепеwаl process, In this section, the complete preventive rесочеry, after feeding а signal, is опlу considered so that А is omitted in the meanwhile. The case of incomplete rесочеry is studied in Section 2 when the death process control is considered.

2,3. Cost fuпсtiоп The quality of functioning of such а system is estimated Ьу the expectation of stationary losses реr unit time:

,

А,:

(со + c|Eпf)pf

* d\m.п.А) +

с"ЕY(0)

,:., (J,

whеrе Е0 is ап expectation of regenerative cycle duration, ЕY(0) is ап expectation of the path integral очеr the cycle 0,

and defined Ьу Jbrфtrll dl; р1 is а probability

the ассumulаtiоп process, Y(t):

Tg,vw

:

* Еmiп(т,Zу,") * рlЕт1 P{Zv,w < т}, and

of failure in one rеgепеrа-

denotes the appropriate

included in Eq. (5).

Note that the passage time frоm state j, j С U, into set ИlУ coincides with the response time of the process in set U if the process has started frоm state j. Therefore, for the distribution function Fi.йt) of the passage time frоm state j into set ИУИ, the MaTkov rепеwаl equations can Ье written [б].

-;

J.

Qt;@s)Fpw(/

-

s)

: : j€чw

Qi;(t),

(6)

It is possible to соmроsе fоr the expectations of passage times, |,уlу : EZi,yy, а linear system of algebraic equations, respectively:

jeU

rф

mi: | / dc,(r), J0 GiQ) : ZQ,;O j€х

ot lп; :

ZrцТ,r,

jex

(8)

i С U.

76,ца, is calculated frоm Eqs. (7) and (8). Introduce the following designations. lw : птпlп,fl >- 1, х" с WV|X(O) с U} is а moment

Непсе,

of the first rеасhiпg set Иl7 from set

jlx(o): l} is

Ц.

fij

: Р{Х(Ь.):

the probability that at the moment of the

first reaching set Y!I/ the process is found in state

j

given

state l.

А liпеаr system of algebraic equations for itiesfi is easily composed:

j€

(4)

('7)

where и;: Ef; is the expectation of response time of X(r) in state j. The following equations hold:

the чаluеs included in Eq. (З), explicitly оr imрliсiф,

depend on раrаmеtеrs m, п алd А. Тhе optimization of the function given Ьу Eq. (3) with respect to these parameters will atlow us to find theiT optimal values:

ieU

Ti,vw-|PiiTl,vw:mi,

fij:Pij+|rй,

failure с9, losses per unit of idle time с1, cost of the control system d(m,п,Ь), and cost of maintaining the unit с". A1l of

min s(и, п, А).

Z9;уц,

(5)

expectation frоm Eq. (1). Let us ascertain the values

tive cycle. The following values аrе known: the cost of

m,п,Ь

*

i€U.

The two-level control policy applied to the degradation process X(r) trапsfоrms it in а controlled regenerative

s(rr, rl, Д)

Е0:

Fi,wQ)

parameters:

. .

7-g,y

2.4. Markov rепеwаl еquаtiопs

After feeding а signal there can Ье two events: either the preventive rесочеry in time т or failure in time Zу,lц w|l| happen. Thus, the two-level policy control depends on thTee

о

As for each realization of X"(r) the equations:

Zv,w : Zo,w, Р{h' : h,wl ) 0, and Eq. (2) аrе fulfilled, the expectation of the regenerative cycle duration therefore can Ье written as

where р7

It is seen iп Fig. 1 that there is а case when

Zv.w : 0,

Thus, the task is to derive algoгithms permitting to calculate Е0 , ЕY(0) and р1 as functions of these parameters.

jeWV, iCU

the probabil(9)

from which the probabilityfi; of the process being at state j, VW, at the moment of feeding а signal is calculated. It is necessary to note that P{Zyy - 0} : Z.wfoi, and this is the probability that the mоmепt of feeding а signal coincides with the moment of fаilurе in this case. As X(r) is а semi-MaTkoy рrосеss and "fo;, j € WV, are initial conditions fоr determination of the function

V.Д. Корпоч / Reliability Епgiпееriпg апd Systern Safety

Fv,w(t) : P{Zv,w ( /}, then using Jordan's thеоrеm of the гepresentation of any probability distribution as а miхturе of atomic and continuous distributions [б] we compose

Fv,wQ): |foiFi,w(t) + zfojq(t) j€w

iЕv where F;,1y(r) :

P{Zi,lry < /}, and Н(r) is а Heavyside func-

tion. Тhеrе is а system of Markov renewal equations for r;ц,(r) similar to Eq. (б):

Fi.g,(t)-

: j€w lГ:u

Qi;@s)F'wG-

s): ZQ,/t), j€w

iCUV from which f';g,(r) is found, and consequently pt and

Е min(T, Zy,y). Тhе рrоЬlеm of calculation of these ча]uеs proceeding from known distributions is considered furйеr in mоrе detail. Тhus, if all these values can Ье ca]culated, then it is possible to find Е0 as а function of раrаmеtеrs и апd п.

If the

preventive recovery of unit's properties was incomplete, then it would Ье required to solve twice the renewal MaTkov equations and apply twice the operation

б

( 1999)

1

-I1

reaching set YlY'up to the moment

ali) "

- Jo[л k-cu : nr,or)at(,

-n

/:

:rfrrG(dr).

And finally, the equation determining form

ЕY(Ц,vw\:

r* оо(r) Jo

ЕY(hй

jе

U.

takes the

dFO,иу(/).

Let us find the expectation of the accumulation process in the interval between feeding а signal оп rесочеry and recovеry itself. Let Y!t) Ье the accumulation process at time r if the process started from state J, and $ mлп(т,Z,,w) ЬУ

:

definition. Тhеп the following equations hold:

t: j€Zt/(lvw ч r1({)

:

: i)

YiZl,ilI(Zl,w < d + \0)I(Zl,w >

т)

(10)

вY(а: |вцG)fоi. jеV

of expectation.

The last equation just defines the rеquirеd чаluе with

2.5, Дссumulаtiоп process

known/9; from Eq. (9).

Сопsidеr the рrоЬlеm of determination of ЕУ(0). As Y(0) is ап additive functional, it is possible to divide the cycle 0 into intervals and consider the values of the functional in these intervals. Let us present Y(0) as У(0)

: y(Ц,ч, * :

as Г(тr)

пiл(т,Zу,у) +

Tgl(T > Zu,"))

Y(Zo,v,w) + Y(miп(т,Zу,у))

:

9.

Consider the accumulation firsfly in the interval[O,h,vw) bearing what is h,учry iп mind. It is the moment of the first reaching of set ИI4lhачiпg the distribution function Fo,vy,tt) defined Ьу Eq. (6).YQо,чi is ап аrеа uпdеr а trajectory of the degradation рrосеss terminating at the moment of reaching set YIU. Then it is сlеаr that all such trajectories mау Ье rеаrrапgеd on the principle of а membership of o-algebra of аррrорriаtе eyents, with which the рrосеss X(r) enters the set аt,h,чуч.Nоw it is necessary to introduce the conditional expectation Е{Y(h1фlЦ.-) and usе the thеоrеm of total probability ЕY (ZO,чil

:

Е@(Y (^,Vw)l^,u"ll.

Hence, we should find

q(t): E(y0)|x(0):j), j eU where / is а moment of reaching set ИIV. Using а method of characteristic function [б] the Маrkоч rепеwа] equations аrе obtained for the expectation of the аrеа uпdеr the process

The рrосеdurе of searching the чаluе

ofEY(() straight Ьу

Eq. (10) is rather awkward, requiTing the repeated solution

of the Маrkоч renewal equations. То find the expectation of the accumulation process оп а trajectory of the process in а regenerative cycle, it is possible to act otherwise. Namely, introduce а concept of accumulated response times of the рrосеss in statej,j С Ц uр to leaving set U оr tГИ, find their expectations from the correspondent renewal equations, integгating with respect to the whole space of states. It should Ье pointed out that though this method, as rаthеr transpffent, will Ье applied in Section 2, but it requiTes again the sequential solution of the rепеwаl equations. Fоr sеmi-Маrkоч nonmonotone processes, in соmmоп case, this рrосеdurе is not so simple. Consider the thiTd method of the determination of ЕУ(0). Ву viTtue of that &(r) is а regenerative process, Smith's thеоrеm [7] give us:

Еу(0)

: F о Г*р{х(r) кТчч J

:/ 0} аrе shown in Fig. 2. As, against the previous чаriапt, is considered, the incomplete rесочеry of properties of the uпit is possible, which is determined Ьу the value А. То exclude а degenerate case, suppose that Ь } m, and besides, Ь,: п - m,Тhе last supposition is not principal and sеrчеs only for diminution of пumЬеr of variables frоm thTee to two when the cost function given Ьу Eq. (3) is optimized. 3.1. Calculatioп of the cost .fuпсtiоп

The expectation ofregenerative cycle durаtiоп Е0 ofthe controlled death process is

:

i:k+l п

where

Е0

I

,.

Е min(T,Z.,g) + P|Zm,o < т}(Ет7 * Tn,-) +P|Z-,o

2

2. Let Fл,g(t): P{Z-,o { r} denote the distribution fuпсtion of transition time, which should Ье found frоm the solution of Eq. (15) with initial conditions given Ьу Eq. (16), where и is taken instead of n, ltm+i : 0 and дr,о : g. The lаttеr is соrrеsропdепt to that the рrосеss starts frоm slate m, and state 0 is absorbing. This mеапs that we аrе interested iп finding the distribution of the moment of the first оссurriпg in state 0 given state m, but not the probability that X(r) is in state 0 at r, Following Ref. [8] and applying the Laplace trапsfоrm, we hаче

Fл,оU\:1- Fl l".i

exp(-tr,til)

ij

Lbia'\- lli)

whеrе

с(х)

:

(х + д1,1)...(.т -| t-c*).

The probability offailure iп one regenerative cyclep; is defined as

рr

:

: I r'Jo

P|Z-,o < т}

,,,n11.1

dC1l).

If G(r) is ап exponentia] distribution with раrаmеtеr the probabiНty offailure is

Ц: mt",H.i i:1

(v 1-

p)p,pl{-

l;,

p,i)

The similar рrосеdurе can Ье applied to determine

PkQ): Р{х(/) : ъlхQ): ml.

т|Ту,-

(17)

апd, P{Z^,K > tl is that, is not an absorbing state, and in the

The difference between Eq, {l7)

where Zr.. is the expectation of the transition time from а casual state у af\er preventive rесоyеry to ýйte и. Other symbols аrе similar to used in the previous section. То determine the expectation of losses реr unit time given Ьу Eq. (З) in the stationary case as а function of parameters m алd п is necessary to obtain the аррrорriаtе formulas and algorithms permitting to express Еrпiп(т,Z^,g),Р|Z-,о = т|, Т n,^, Ту,-, ЕY (0) thTough these paTameters.

l.Let {о,k:l,...п,

denote the response time of the process in state ft. As X(r) is а Маrkоч death process, ýд has the exponential distribution function Ll,Q) :

ехр(-д,lr) and expectation Е{1,: expectation of transition time

I

Тдо,1

Il

ра,. Then the

frоm state

/.0

to state

in the first case,

/с

second is, i.e. дl,о : g. The probability of staying X(r) in state staTted frоm state и at 0, is

pkl):

m

H**t.,.H*

/с

at r, if X(r) has

ехР(-р")

i i:K+t triaL+l(-r.c;)

whеrе

ot+t(r)

-

(х

* rч,+)...(х -|

t_b-)

conditional expectation of the transition of the controlled process frоm а саsuаl state у(т), where X.Q) has оссurrеd after preventive rесочеry, to state ,fl,

J. LetTylTl,mbe а

V.А. Корпоч / Reliability Епgiпееriпg апd System Safety 66 (1999) 1-11

t.e. Tyg1,-:B(Zrg1,^|T). Тhеп ВТуgl,m.

Ъ,. is

defined as Ту,m:

Непсе

!-

Туtт\.m: )

i:A+

:

Then Тr,-

3.2. Asymptotic reliability fuпсtiоп

Т;.,Рl_ц(т)1

JГ Ъt,l,. dG(/). If G(r) is exponential with

раrаmеtеr у, then

Ту,-: Ё i:Д+1

z ь (" vРt-л+t"'lrmt

TL.

+ tф t-ri.'

i__?

t"i)

4. Using the fоrmulа fоr the expectation of а miпimum

of

two independent variables and the Laplace trапsfоrm, in case of exponential G(r), the next чаriаЬlе is found as

Еmiп(т.Z.п): " -lll'v' 5.

п

,1 (" +

'

11,1\1l";tol(-

1l.i)'

If to consider that the expenditures on maintaining the uпit at а state depend linearly on this state, then the finding ot ЕY(0) is rеduсеd to the calculation of some expected чаluеs of the controlled process in one regenerative cycle. Let &ь k : 0, -.., п, Ье а random time frоm 9 When X"(t) : k.Fоr the sйtionary case, Еy(0) is deter_

mined as

Еу(0): iou*n.

(18)

Е0

The expectations of the variables from Eq. (18)

аrе

prB{r + p"f(E("f +

ЕО,

pt *

рпr

:

where рб is the probability of nonfailure in the cycle. Applying the Laplace transformations to Eq. (19) and the

Taylor-series expansion of the Laplace transformations of the correspondent variables in the neighborhood of zero, it is possible to obtain the approximate equation:

р{(=/}

-*n(-#)

tion with arbitrary р1 the inverse Laplace transformation should Ье applied to Eq. (19), though it is not а simple

'r\

- F--i"'Fr-tП-t '-"' -l p,)p,l,to'^_,F tй о*_, (v )' '

i:O,...,m-|

*P,-ApT;bk),

i:0,...,m -

of

corrosioп

v* P-*t

E*n-;:

(19)

I

3.3. Numerical simulatioп of the controlled process

E8-_i

Еф:

Et:

computation task.

: рlЕт1

"(,

Additionally, such ап important chaTacteristic of ореrаtion quality as the reliability function of а unit maintained on the principle of two-leyel policies is also of interest. Assume now that the failure is а rаrе event, i.e. the рrоЬability offailure р1 in one regenerative cycle is suffrciently small. Then it is possible to find the asymptotic reliability function taking advantage of the Laplace transform for distribution functions. Let (1 Ье the time fгоm the beginning of а rеgепеrаtiче cycle to failure if it was with failure; ("r Ье the duration of а regenerative cycle without failure; ( Ье the time from the beginning of а cycle to the first failure of the unit. Then it is possible to write

when р1 is of small чаluе. То determine the reliability func-

defined as Et}g

Thus, all values for calculating Eq. (З) can Ье found as functions of раrаmеtеrs of the control policy.

2

1, pi

It was assumed hеrе again that the time of preventive rесочеry is exponential. This is essential only in that this permits

to calculate the variables from Eq. (3) in explicit way. Рrосеdurеs of numerical integration of imрrореr converging integrals will Ье required otherwise.

The algorithms derived еаrliеr cannot Ье realized practi-

cally Ьу means of analytical expressions. It is necessary

thеrеfоrе to usе пumеriсаl methods for evaluating cost fuпсtions when optimization procedures аrе implemented. Some results of numerical simulation of the corrosion process,

controlled Ьу the two-level policy, are accounted further to verify these algorithms. Consider

а

model of control in case of where а mаjоr unit

is subjected to aggressive exterior influence. Аssumе that

this element has а corrosion-resistant сочеr of width й, : пй, whеrе Л is а unit of mеаsurеmепt of corrosion losses of the lауеr. The mеаsurе of losses is possible to estimate either Ьу subtraction frоm initial thickness (thinning) or Ьу а gravimetric method. When the layer rеасhеs zеrо чаluе, the failure of the unit оссцrs. Let а device Ье there signalizing аЬоut forthcoming failure when the lауеr reaches some level h*: mЙ. Дftеr feeding а signal, there can Ье two events: either preventive recovery of чаluе АЙ, А : п - m, in time т, оr failure if the rесочеry was late. After the failure, we have to restore not only the unit

V.А. Корпоч / Reliability Епgiпееriпg апd System Safety 66 (1999) 1-11

8

ТаЬlе

stationary Markov chain

1

Cost factors for simulation of Eq. (3) do

С1

with rапgе space Х -

{О,1,,.,..,rп,,..,п...|. The process starts frоm state 0, and state п is for failure. The previous account could Ье easily

d1

modified for the chain as in Ref. [3]. Ноwечеr, the statement

1

l

1

0.1_100

1

2

1

0.1_100

1

1

з

0.1_100

1

4

10

l

1

1

l0

of the рrоЬlеm will differ frоm the previous sections. Namely, the uпit is unique апd recovery is impossible.

Replacement of the failed unit is a]so impossible or it is а rare event. After feeding а signal, when state и was reached, the single operation to prevent failure ofthe unit is stopping.

1-1ш

but possibly the whole system. Thus, the cost of accident repair с€ш Ье of significant value.

Assume further that the colTosion process can

After а delay т the stopping is fulfilled. If the stopping is

late, the unit rеасhеs state п апd fails. call this case а unit of single use. As before we аrе interested in finding the optimal

Ье

values of m ald п. The stationary cost function given Ьу Eq. (З) is not аррrорriаtе, because to соuпt on а large

described Ьу а MaTkov death process and the control policy is applied to it as. was considered earlier. So it is required to determine values,?, and п giving а miпimum to Eq. (З). Following Ref. [9], it is accepted that the greatest possible yаluе,х is по mоrе than 11, and rп varies frоm 1 to 10. The death рrосеss is stationary, and ;r,; : 1,2 mопй-l fоr апу l. The preventive recovery ofthe covering is саrriеd оut with а delay т having an exponential distribution. The cost ofthe control system includes the cost of request for rесочеry, being а linear function of the rесочеry value, d(A) : й + dlA. The cost factors conИined in Eq. (3), which were used fоr simulation, аrе listed in Table 1 (ranges of numbers in some cells mеап batch variations). The simulation results, as points on plain, corresponding to optimal m апdп are shown in Fig. З. It is seen that the diminution of both notifying levels follows the growth of the failure cost and time of

number of replacements, tending to infinity, is not rеаsопable. То obtain mоrе explicit results the cost function will Ье simplified. Such а statement of the control рrоЬlеm fоr а single unit differs frоm others, fоr example, when the probability of fаiluге is minimized оr а minimax рrоЬlеm with constraints is solved. Ноwечеr, this mау Ье considered as the first esti-

mation fоr optimal degTadation values and maintenance costs when а simple stopping is implemented. It mау Ье used instead of an optimal stopping procedure fоr the unit equipped with а monitoring system evaluating sequentially

сumulаtiче damage and expected cost

of

stopping as

described, for ехаmрlе, in Ref. [4], and considered as а degenerate case of an optimal stopping procedure when there is only one moment for decision making.

fесочеry. The decreasing of the signal level (curve 1) follows the increasing of the cost of request. The large cost се ofholding the controlled рrосеss uр at а level significantly affects the fаilurе state (сurче 2). Thus, we mау

4.1. Cost fuпсtiопs

Fоr а unit of single use, let the cost function Ье

conclude that optimization is required not only for the signal level but also for the failure level, which was not so obvious from the beginning.

s{m,п):"(?!ЦР)

4. Маrkоч chain

where 1 : I{Z-.n < т} is an indicator, с9 is а cost of stopping, and с1 is ап additional cost if failure оссurs. Let us write Eq. (20) as

Let the degTadation process of а unit Ье described Ьу а

S1@l,п): соЕ(0-1) + с,Е(10-1)

!) Ф d d

о о

1

optimal signal level

Fig. 3. Modeling of corrosion: optimal values of

п

и

and z for Eq. (4) with cost factors frоm ТаЬlе 1.

trol

(2I)

V.Д. Корпоч / Reliability Епgiпееiпg апd System Safety 66 (1999) 1-11

and consider in mоrе detail the second tеrm in Eq. (21). The

stopping moment miп(т,Z^,).

0 сап Ье

written

as 0:

Zg,-

*

D: |D^,...,Di,...l,i>

m, Ье а partition of the where D;: {а!0: j}. Let Sл: a{D-,...,Di,...} Ье o-algebra formed Ьу an accounting sequence of these events. Clearly, 0-1 is Sr-measurable

Let

О,

elementary space

random variable and can Ье written as

а-':

ф

I

i:m

Е(Е(10-1 lSD))

and finally

Е(10-|

)

: > i-'E(I i:m

where и is ап absorbing state and the аррrорriаtе element of the transition probability matrix is such that р-*: 1. It is clear that Eq. (23) is the probability that the time of the first

Ф

а

i-m

i:m

i:m

j:0

jlP{Z_."

:

'} In the simplest case considered hеrе, we mау act other-

i

:r} : I P{Zo,_ : i - j||р{Zл,, : j:0

Р

j}P{T = j}

Let us write the expected cost per uпit time fоr the unit of multiple use, when multiple replacements аrе possible

с -l dP{Z^n < т|

(22)

Е0

and соmраrе further the results of modeling with Eqs. (20) апd (22),

4.2. Еlеmепtаry Markov сhаiп

Consider а degradation process modeled Ьу а mопоtопе homogeneous stationaTy MaTkov chain with unit jumps. The transition probabilities do not depend оп а level. At discrete moments of time i : |,2..., with probabiliýp the chain сал jump up to the next state, and with probability q the state

р+ q:1. Let the delay т have

То find

Рlh^: i|,P{Z^":

l.

lh- :

i:m,m

Thus, the main formulas for calculating Eq. (20) for each

Poisson distribution with parameter

:0,

1,..., i.

Binomial distribution

mалdп are obtained.

rеmаiпs the same,

\,2,..., j

the trajectories contained in event {Щ*: f } should Ье examined. These аrе trajectories which somehow wеге in state m - l at time j - 1 апd entered state m at i, Тhе probability of tЫs event is distributed Ьу the negative

+Р{т: j}P{Z^,> jl7.

s2(m,п):

:

(',)on'-',

То find the distribution of time of the first reaching of state

where

р{0

:

и

j}P{T ="l}

i-m

i.e.

wise, taking advantage of the interpretation of Р{Х(r) : j|X(O):0} in terms of Bernoulli trails. Then the event {X(r):j|x(0):0} should Ье considered as j positive results fоr j trails:7 < j, апd

ptx(,): j|x(o):0}

lZ-.n < т}l(Dl)).

The last equation shows the way of calculating Eq. (21). Thus, Eq. (20) is determined as

: i_

(2з)

Р{^^<

probability:

:

Ptx(r): иlх(0):0}

entering state m is less оr equal i, : Р{х('): -lx(O):0}.

i-|t1o,y.

Fоr Е(10-1), it is possible to write the thеоrеm of total Е(10-1)

necessary to solve, in general, forward balance equations similar to Eq. (9), and as was realized in Ref. [З]. Fоr а monotone chain, the method of generating functions is quite suitable, and the following probability can Ье found:

the

j} and other probabilities

for а homogeneous stationary Markov chain with а matrix of transition probabilities of ап arbitrary fоrm, it is

U

:

:

(*-_' r)о'

n'-^,

*

:

|,2,..,,

+ 1,...

These derivations аrе sufficient to determine all values included Eqs. (20) апd (22). Fоr convenience, all of the necessary formulas аrе written as follows. Their direct derivations, owing to theiT evidence, аrе omitted. Fоr the unit of single use

Рlh*: i -.i|

-i-1\ " :{(, m-| l|pпql-J-m. i-j=-, 0, Oп-m, : {(, ',),,-,о-"-^, ^'0п-trl 0