Periodic replacement when minimal repair costs depend on the age and the number of minimal repairs for a multi-unit system

Microelectron.Reliab.,Vol. 30, No. 4, pp. 713-718, 1990. Printed in Great Britain.

0026-2714/9053.00 + .00 © 1990 Pergamon Press pie

PERIODIC REPLACEMENT WHEN MINIMAL REPAIR COSTS DEPEND ON THE AGE AND THE NUMBER OF MINIMAL

REPAIRS FOR A MULTI-UNIT SYSTEM SHEY-HUEI SHEU Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, Republic of China

(Receivedfor publication 4 July 1989) AImtract--A policy of periodic replacement with minimal repair at failure is considered for a multi-unit system which has a specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed for any intervening component failure. The cost of a minimal repair to the component is assumed to be a function ofits age and the number of minimal repairs. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. The necessary and sufficient conditions for the existence of an optimal replacement interval are found. 1. INTRODUCTION

2. ONE-UNIT SYSTEM

In most reliability models of repair, it is typically assumed that the repaired part is returned to a "good as new" state (perfect repair). This assumption means that upon repair, an item has the same life distribution as when it was new. Although this assumption provides technical convenience, in many situations it is not realistic. A more realistic situation is that when the item is repaired, it is restored to its functioning condition just prior to failure. This is called minimal repair. That is, if the original life distribution of the item when it was brand new was F, then the item upon repair may have survival function ,v where t is its age at failure and Pt(x) = l~(t + x)/P(t). Barlow and Hunter [1] considered the case of periodic replacement of overhaul at times T, 2T, 37", . . . . (for some T > 0) and minimal repair if the system failed otherwise. They considered the cost of replacement, c2, and also the cost of each minimal repair, c~. In their paper, Barlow and Hunter show how to calculate the optimal period T assuming the cost of a minimal repair is constant and using as an optimality criterion the minimization of total expected cost per unit time over an infinite time horizon. Boland and Proschan [12] generalize the Barlow-Hunter model to incorporate the situation when the cost of a minimal repair is an increasing function of the number of previous repairs to the system. Furthermore, Boland [11] considered that the cost of a minimal repair to the system which fails at age t is C(t), where C(t) is a continuous non-decreasing function of t. Hence, as the system ages it becomes more expensive to perform minimal repair. In this paper we generalize the Barlow-Hunter model to incorporate the situation when the cost of a minimal repair depends on the age and the number of minimal repairs to the system. In Section 2 a one-unit system is discussed and in Section 3 a multi-unit system is discussed.

In this paper, F denotes the life distribution function of the system with density f, ~.(t) denotes the failure rate function of the system, while A (t) = S~A(s) (:Is is the cumulative failure rate function or hazard function of the system. We consider a periodic replacement when minimal repair costs depend on the age and the number of minimal repairs for the one-unit system according to the following model.

Model 1 An operating system is completely replaced when it reaches age T(T > 0) at a cost co. If it fails at age y < T, it undergoes minimal repair. The cost of the ruth minimal repair is Cm(y). After a complete replacement the procedure is repeated. We assume that all failures are instantly detected and repaired. Let N(t) denote the number of minimal repairs performed on the system of age t for t in the age internal [0, T). We know that {N(t); t >tt 0} has a non-homogeneous Poisson distribution with parameter A (t) (see Barlow and Proschan [2], pp. 96-97). If C(T) is the total cost of minimal repair in [0, T), the following theorem gives a useful expression for the expected costs of minimal repair in [0, T).

Theorem 1 The expected minimal repair cost of the system in the interval [0, T) is

E[C(T)I = [Zh(y)A(y) dy jo where

h (y) = E [CN(,+l(Y)]Proof. If Sl, $2. . . . . SN(z) are the times of the minimal repairs, then the total minimal repair cost in 713

714

SHEY-HUEI SHEU

the interval [0, T) is N(T)

Ci(S3.

is non-decreasing continuous and hence C'(T) has a zero if

{h(T),~(r) -- h(y)A(y)}

lim

i=l

T~oo

dy > Co.

Hence, the expected minimal repair cost is Under the strictly increasing assumption, C(T) is strictly increasing so To is unique.

[-N(r)

Remark I

= ~ Etc,(s,); s, < r] =

C~(y)p(Si~dy ).

If Cj(y) increases in j and y, then h(y) increases. Also, if Cj(y) is continuous and h(y) is finite for all y > 0, then h(y) is continuous.

i=l

But

3. MULTI.UNIT SYSTEM

P(S, <~y) = P ( N ( y ) >11i) = ~ e -a(y).Ak(y)/k! k=i

and so ~ P ( S ~ d y ) = e - a ( Y ) Ai-~(1~ (i. 2(y)dy

= P ( N ( y ) = i - 1)2(y) dy. Thus

e[c(r)] =

C , ( y ) . P ( N ( y ) = i - 1)2 (y) dy

In this section we discuss periodic replacement when minimal repair costs depend on age and the number of minimal repairs for the multi-unit system which has a specific multivariate distribution. To characterize the model more explicitly we first define the specific multivariate distribution. Let F be a class of univariate distribution (e.g. F = {exp} or F = {IFR}). A random vector X = (Xl, X2. . . . . X,) or its distribution is said to be in M V ( F ) if there exist independent random variables

i=l

Ui.,,FteF

= j ~ E[CN(y) + 1(y)]2 (y) dy. Let us now consider the problem of finding a period To for replacement which minimizes the expected long-run cost per unit of time. Co denotes the cost of a planned system replacement. If ~'(T) represents the expected long-run cost per unit of time when the system is periodically replaced at times T, 2T, 3T, . . . . then ~'(T) has the form

~(T) = {forh(y)2(y) dy + co}/ T.

Uij~FiyeF

for i = 1,2 . . . . ,n, for l<~i
for

1 ~< i < j < k ~
UI,2,3. . . . . . ~ FI,2,3...... e F

such that the survival function of X = (X~,X~ . . . . . X.) is given by /~(X 1, X2 . . . . .

Xn)

= P ( X 1 > x i ) X2 > x2 . . . . .

Therefore

Xn>xn)

= P ( U j > x . 1 <~i <~n; U~j h (T)2 ( T ) T - f ~ h (y)2 (y) dy - Co C'(T) = dOT2

> max(x, xj). 1 ~ max(x1, x2 . . . . . x.))

and this yields the following theorem. = f i F,(x,)

Theorem 2

i=l

If h ( y ) 2 ( y ) is a non-decreasing continuous function of y, then an optimal replacement To exists. To is finite if lim T~

j.r{h(T),~(T)

-- h ( y ) 2 ( y ) } dy > Co.

0

Furthermore, if h ( y ) A ( y ) is a strictly increasing continuous function of y, then To is unique. Proof If h (y)2 (y) is a non-decreasing continuous function, then

h ( T ) A ( T ) T - f r h ( y ) A ( y ) dy

Jo

]-]

P,j (max(x/, xj))...

l~i
rj.2.3...... (max(xl, x2 . . . . . x,)) for xi_> 0,

i=l,2,...,n.

If F = {exp}, then MV(exp) is the class of multivariate exponential distributions as defined by Marshall and Olkin [15]. In the bivariate case, if X = (X 1, X2) is in MV(F), then there exist independent random variables U~, U2, U~2 each with distribution Fl, F2, Fl2 in F satisfying

( X ' 2 ) = ( m i n ( U , , Uj2)N~ \min(U2, Ul2)]

Minimal repair costs for a multi-unit system and the survival function of X ffi (X~, ,i"2) is given by R(x~, x:) = P(XI > x~, X2 > x:)

ffi P(UI > xl, U~ > x~, U~ > max(x~, x~)) = / v I(x~)F~(x~)F~2(max(xl, x~)). To fix ideas, we first consider the following model for the two-unit system which has the above bivariate distribution.

715

Theorem 3 The expected minimal repair cost of the two,unit system in the interval [0, T) is

E[C(T)]

~r° [ht(Y)Al(Y) + h2(y)22(y) + h~2(y)Al~(y)] dy,

where

Model 2

hi(y) = E[CLN,ly)+ I(Y)],

A two-unit system starts to function at (the same) time 0. Unit 1 contains components C~ and C~. Unit 2 contains components C2 and C~. Component CI (respectively (72, C~2) has life distribution FI (respectively F,, FI:). The two-unit system is completely replaced whenever it reaches age T ( T > 0) at a cost co. If component C~ fails at age y < T, it causes the failure of unit 1 and undergoes minimal repair. The cost of the ruth minimal repair is C~,,,(y). If component C: fails at age y < T, it causes the failure of unit 2 and undergoes minimal repair. The cost of the ruth minimal repair is C~,~(y). If component Ct~ fails at

h~(y) ffi E[C~.N2~y)+l(Y)] and

ht2(Y) = E[Cm ~',2('~ + I(Y)]. Proof The result follows from a similar argument to theorem 1. Let us now consider the problem of finding a period To for replacement which minimizes the ex, pected long-run cost per unit of time. If C(T) represents the expected long-run cost per unit of time when the system is periodically replaced at times T, 2T, 3T, . . . . then C(T) has the form

~r Jo__[hl(Y)Al(Y) + h2(Y)22(Y) + ht2(Y)Al2(Y)] dy + co C(T)

T

Therefore

~'(T) = [hl(T)21(T) + h2(T)22(T) + hI2(T)212(T)]T T2 age y < T, it causes the failure of both units and undergoes minimal repair. The cost of the m th minimal repair is C12.,,(y). After a complete replacement, the procedure is repeated. Let 21(0 (respectively 22(t), 2(2(t)) denote the failure rate function of FI (respectively F2, F12). Let Nl(t ) (respectively N2(t ), N12(t)) denote the number of minimal repairs performed on the component C~ (respectively C2, C~2) of age t for t in the age interval [0, T). We know that the counting process {Nl(t); t >//0} (respectively {N2(t); t 11>0}, {N12(t); t t>I0) has a non-homogeneous Poisson distribution with parameter l

Al(t) = [ ' 21(s)

ro [hl(y)21(y ) + h2(y)A2(y ) + h12(y)212(y)] dy + Co

ii and this yields the following theorem.

Theorem 4 If 2

~. h~(y)2,(y) + hl2(y)Al2(y) i-I

is a non-decreasing continuous function of y, then an optimal replacement interval To exists. To is finite if lim

T~oO ~0 ~.i=1

h,(r)2,(r) + hl,(r)2,2(T)

-- ,=I ~ h,(y)2,(y) - h12(y)lq2(y)}dy > co. Furthermore, if

do

2

respectively

~. hi(y)21(Y) + ht2(Y)~.I2(Y) i-I

A2(t) = In 22(S) ds, Al2(t ) = []

~I-12(3)

ds.

i J v

If C(T) is the total cost of minimal repair of the two-unit system in [0, T), the following theorem gives a useful expression for the expected costs of C(T).

is a strictly increasing continuous function of y, then

To is unique. Proof The result follows from a similar argument to theorem 2. N o w we consider the following model for the n-unit system which has the above multivariate distribution.

716

SH~Y-HUE~ SI~u

Model 3

respectively

The n-unit system starts to function at (the same) time, 0. The N-unit system contains components C~, i = 1 , 2 . . . . . n; C U, l<<.i 0) at a cost Co. If component C, fails at age y < T, it causes the failure of unit i and undergoes minimal repair• The cost of the ruth minimal repair is C~,.(y). If component C u fails at age y < T, it causes the failure of units i and j and undergoes minimal repair. The cost of the ruth minimal repair is C~j..,(y). If the component Cuk fails at age y < T, it causes the failure of units i, j and k and undergoes minimal repair. The cost of the ruth minimal repair is Cok,.(y ). Similarly if the components C~,2,3...... fail, it causes the failure of units 1, 2, 3 . . . . , n and undergoes minimal repair• The cost of the ruth minimal repair is C~,2,3........ (y). After a complete replacement, the procedure is repeated. Let 2~(t) (respectively 20( 0, ).~ik(t). . . . . )-~2...... (t)) denote the failure rate function of F~ (respectively F~, Fot . . . . . F~,2,~...... ). Let N~(t) (respectively Nu(t), Nqk(t) ..... Nt,2, 3...... (t)) denote the number of minimal repairs performed on the component C~ (respectively Cu, C~j~. . . . . C~, 2,3...... ) of age t for t in the age interval [0, T). We know that the counting process {N~(t); / ~ 0 } (respectively {Nu(t); t ~O }, {Nq~(t); t ~ 0 } . . . . . {N~,2,3...... (/); t ~ 0 } has a non-homogenous Poisson distribution with parameter

A°(t) = J0 2u(s) ds, Aok(t) = ;o 2qk(s) ds, AI,2,3 ...... ( t ) - ~

'~1,2.3. . . . . . ( s ) d s .

If C(T) is the total cost of minimal repair of the n-unit system in [0, T), the following theorem gives a useful expression for the expected costs of C(T)•

Theorem 5 The expected minimal repair cost of the n-unit system in the interval [0, T) is f'rF " h

2

L,Z -),.)+ "ffi

+ ~

i
hqk(y)A,/k(y)+ "'"

i
• .. + hl,2,3 ...... (Y)21,2,3 ...... ( Y ) / d Y d

where

hj(y)=E[Ci, N:y)+l(y)] for i = 1,2 . . . . . n, hq(y)=E[C~j, No~y)+l(y)] for 1 ~< i < j ~
= E[CIY,~.3....... u,.:,,...... ~,)+ l]Proof. The result follows from a similar argument to theorem 1. Let us now consider the problem of finding a period To for replacement which minimizes the expected long-run cost per unit of time. If C(T) represents the expected long-run cost per unit of time when the system is periodically replaced at times T, 2T, 3T, then

A,(t) = I' 2,(s) ds J0

t'rl- . h

fo

2

~'(T)-

'
--I

Y'. huk(Y)J.l~(Y) + ' ' " + hl,2,3 ...... (Y)21,2,3 ...... ( Y ) / d y + C0 J

i
T Therefore

~"(T) =

[~

h~(T)A,(T)

,-1

+ Z hu(r),b(r) + ~ h,:~(70~.~(r) ,
ht 2 3

T2

,
.(T)2~ 2 3 .,.(T) T -

h,(y)R~(y) + ~ h~j(y)Ao(y ) ,)0

...

+

+""

li--I

+ •..

i
T 2

1

hqk(y)Rt/,(y) + "'" + hl,2,3 ...... (Y)'~1,2.3 ...... • • , _~_ t < j < k

T2

(y)/dy d

C0

and this yields the following theorem.

Minimal repair costs for a multi-unit system

Theorem 6 If

h,(y)21(y) + ~ ho(y)2o(y) i-I

717

[toil + 1, to~l]. Therefore, if g(t) is increasing and continuous then the optimal replacement interval To is an element in the set {to, to~2, to~3. . . . }. Example. Let us consider the bivariate case

i
( ~ i ) = fmin(U1, U3)'~

\min(U~, u0/

+ X h,~(y)~.,j~+ ... t
" ' " + hl.2.3 . . . . . . (Y)'~1.2.3 . . . . . . (Y)

where Ui for i = 1, 2, 3 has distribution Fi, failure rate function r~(t) and hazard function

is a non-decreasing continuous function of y, then an optimal replacement interval To exists. To is finite if

R~(t ) =

.f~

ri(s )ds.

If

h,(T)2,(T)+ Z h,AT)a,AT)

lim T~°° dO

l.l=l

i
+ ~

h~jk(T)2~,(T) + . . .

C,.m(t) = ci.m+ Ci(t) where

i
c~.m=a~+mci for i = l, 2, 3; m = 1 , 2 , 3 . . . . .

" ' " + hl.2.3 . . . . . . (T)21.2.3 . . . . . . (T)

- ~. h,(y)2,(y)- ~ ho(y)2u(y ) i~l

-

l
~

then

f~ 2 ~i=2~1(aiRi(t)+ GRi(T) + ~Ci Rj(T)

~'(T)

h~#(y)2~#(y) . . . . + ~fro

i
....

h1.2.3 . . . . . . (Y),~.1.2.3 . . . . . . ( y ) ) d y > c o.

If the life distribution is Weibull of the form

F~(t) = 1 - e -ca'°p' and Ct(t) = c~(R~(t))% then the optimal period To must satisfy

Furthermore. if

hi(y)A,(y) + ~ ho(y)Ae(y) i- 1

i
3

3

(a, + c,)(2,ro)P'(fl,- 1) + ~ c,(2,To)2a'([3,- 1/2) ill

+ ~ ht~(y)2~(y)+"" i
i-I

+

iffil

c~/~,(~.,ro)p'~'÷~ l

~ ( ~ + l) =co.

"'" + hl.2.3. . . . . . (Y),~.1,2.3...... (Y) is a strictly increasing continuous function of y, then To is unique. Proof The result follows from a similar argument to theorem 2.

Remark 2 We may also wish to consider the problem of finding a replacement interval To which minimizes the total expected costs over a finite time horizon [0, to). F o r period T, where I T < t o < ( l + l ) T for some integer l, we see that the total expected cost in [0, to) is T

C'°(T)=lc°+I fo g(t)dt + fo°-trg(t)d, where

g(t) = ~ h,(t)2,ft) + ~ hq(t)2e(t ) i-- 1

+ ~

I
hl~(t)gl~(t)+""

t
" ' " + hl.2.3 . . . . . . (I)21.2.3 . . . . . . (t).

If g(t) is a continuous increasing function of t, it follows that C~,(T)=l[g(T)-g(to-lT)]~O in MIt ~/,¢--G

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