A generalized model for determining optimal number of minimal repairs before replacement

38

European Journal of Operational Research 69 (1993) 38-49 North-Holland

Theory and Methodology

A generalized model for determining optimal number of minimal repairs before replacement Shey-Huei Sheu

Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, ROC Received March 1990; revised August 1991

Abstract: A generalized model for determining optimal number of minimal repairs before replacement is introduced which incorporates minimal repair, replacement, and general random repair cost. The expected cost rate is obtained. It is shown that the optimal number n* which minimizes the cost rate is given by a unique solution of the equation under certain conditions. Various special cases are considered. Keywords: Maintenance; Reliability; Systems

1. Introduction

Makabe and Morimura [6,7,8] proposed the new replacement where a unit is replaced at the n-th failure, and discussed the optimum policy. This model has been generalized by Morimura [9], Nakagawa [10], Nakagawa and Kowada [11], and Park [12,13]. This is a modification of the periodic replacement model introduced by Barlow and Hunter [1], in which a unit is replaced at time T and undergoes only minimal repair at failures between periodic replacements. These policies are commonly used with complex systems such as computers, airplanes, and large motors. In this article a generalized model for determining the optimal number of minimal repairs before replacement is presented which incorporates minimal repair, replacement, and costs of the i-th minimal repair at age z which depend on the random part C(z) and the deterministic part ci(z). The policy is described explicitly at the beginning of the next section. We derive the expected cost rate from this model and obtain the optimal number n* to minimize the cost rate under certain conditions. As special cases, many of the results of Makabe and Morimura [6,7,8], Nakagawa [10], and Park [12,13] are obtained. In Section 2 the model is described, then the expected cost rate is found Theorem 1 gives a general optimization result. In Section 3 various special cases are detailed. In the last section a numerical example is given. Correspondence to: S.-H. Sheu, Department of Industrial Management, National Taiwan Institute of Technology, 43 Keelung Road, Section 4, Taipei, Taiwan 10772, ROC.

0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

S.H. Sheu / Determining optimal number of minimal repairs

39

2. General model

We consider a preventive maintenance model in which minimal repair or replacement takes place according to the following scheme. A unit has two types of failures when it fails at age z. Type 1 failure occurs with probability q(z) and is corrected with minimal repair, whereas type 2 failure occurs with probability p(z) = 1 - q(z) and a unit has to be replaced at a cost c 2. If the n-th type 1 failure occurs before type 2 failure, then a unit is replaced preventively at a cost c 2. The cost of the i-th minimal repair at age z is g(C(z), ci(z)) where C(z) is the age-dependent random part, ci(z) is the deterministic part which depends on the age and the number of the minimal repair, and g is a positive nondecreasing continuous function. Suppose that the random part C(z) at age z has distribution Kz(x), density function kz(x) and finite mean E[C(z)]. After a replacement the procedure is repeated. We assume all failures are instantly detected and repaired. We always assume c 2 > 0. Assume that the unit has a failure time distribution F(x) with finite mean u and has a density f(x). Then, the failure rate (or the hazard rate) is r(x)=f(x)/F(x) and the cumulative hazard is R(x)= f~r(y) dy, which has a relation if(x) = exp{-R(x)}, where if(x) = 1 - F(x). It is further assumed that the failure rate r(x) is continuous, monotone increasing, and remains undisturbed by minimal repair. If n = 0% the survival distribution of the time between successive type 2 failure is given by

See Block et al. [4] for a derivation of this result. Let Yl, Y2. . . . be i.i.d, random variables with survival distribution ffp and Y,.* denote the length of the i-th successive replacement cycle for i = 1, 2 . . . . . Let R* denote the operational cost over the renewal interval Y~*. Thus {(Y~*, R*)} constitutes a renewal reward process. If D(t) denotes the expected cost of operating the unit over the time interval [0, t ], then it is well-known that lim t-~

D( t ) t

-

E[ R? ] E[YI* ]

.

(2)

(See, e.g., Ross [14], p. 52].) We shall denote the right-hand side of (2) by B(n; p). We now give a derivation of the expression for E[R~] and E[Y,*]. First, however, we must describe in more detail the failure process which governs the cost over the interval [0, YI* ]Consider a nonhomogeneous Poisson process {N(t), t _> 0} with intensity r(t) and successive arrival times S,, S 2, . . . . At time S n we flip a coin. We designate the outcome by Z n which takes the value one (head) with probability p(Sn) and the value zero (tail) with probability q(Sn). Let N(t)

L(t) = ~ Z n

and

M(t)=U(t)-L(t).

n == 1

Then it can be shown that the processes {L(t), t > 0} and {(M(t), t > 0} are independent nonhomogeneous Poisson processes with respective intensities p(t)r(t) and q(t)r(t) (see, e.g., Savits [15]). This is similar to the classical decomposition of a poisson process for constant p. For our models we have the identification )11 = inf{t > 0: L(t) = 1}. Thus, we also can get the survival distribution of the time until the first type 2 failure is

P(Y1

>Y)=

P(L(Y)=0)=

exp{-

foP(X)r(x)dx}=

ffp(y).

(3)

Given that a type 2 failure does not occur in y, the conditional probability of k type 1 failures during [0, y] is

Pk(Y) =P(M(Y) = k lL(y) = O) = P ( M ( y ) =k) exp(- foq(X)r(x) d x ) ( f o q ( X ) r ( x ) dx) =

k!

k (4)

S.H. Sheu / Determining optimal number o f minimal repairs

40

Therefore, the survival distribution of YI* (the time until the first replacement) is given by n-1

G(Y)=P(Yx*>Y)=

n-1

]~P(L(y)=0,

M(y)=k)=

E P(L(y)=O)P(M(y)=k) k=O

k=0 n-1

= Y'. F.(Y)Pk(Y)-

(5)

k=O

The p.d.f, of YI* is dG(y)

g(Y)

n-1

d~

= E k=O

ff,(y)p(y)r(y)p~(y) +ff,(Y)Pn-I(Y)q(y)r(Y).

(6)

From (5), the expected length of a replacement cycle is E[YI* ]

:i;

yg(y) dy =

/o=V"(y)dy:

n-1

ov

fo Fp(Y)Pk(Y) k=O Y'.

(7)

dy.

We are now ready to derive the expression for E[R~]. A system has two types of failure. A system is replaced upon the type 2 failure or the n-th type 1 failure, whichever comes first. If the i-th type 1 failure occurs before the type 2 failure for i = 1, 2 . . . . . n - 1, then the minimal repair will be performed. Thus the operational cost R~' over the renewal interval Yl* will be n-1

R~ = c 2 + Y'~ g(C(Si),

ci(Si))Fp(Si)

i=1

where S i is the time of the i-th minimal repair and Fp(Si) is the probability that a type 2 failure does not occur up to time S i. Therefore, the total expected cost in a replacement cycle is

E[R~] =c2+E

]

g(C(Si), ci(Si))Fp(Si)

i n-i

=c2+ E fo Ec(z)[g(C(z)' ci(z))]PP(z)P(Siedz) i=l

where Ec(;)[g(C(z), ci(z))] is the expectation of a random variable C(z). But ~

exp

-

P(Si<_z ) = P ( M ( z ) >i) = ~.,

q(x)r(x)

dx

q(x)r(x)

dx

k!

k=i

and so

.z° ex.( joZq(.,r(x,

P(S i e d z ) =

eo( ( i - 1)! = p i _ l ( z ) q ( z ) r ( z ) dz.

.x) q ( z ) r ( z ) dz

S.H. Sheu / Determiningoptimalnumberof minimalrepairs

41

Thus, n--1

E[R~]=c, + ~_~ fo Ec(z)[g(C(z)'ci(z))]FP(Z)Pi-l(z)q(z)r(z) dz i=1 n-I

= ~ + E fo°~hi(z)p~( z)pi_,(z)q(z)r(z ) dz,

(8)

i=1

where

hi(z)= Ec(z)[g(C(z), ci(z))].

Remark 1. We only consider cost structures where hi(z) = Ec(z)[g(C(z), cs(z))] is finite for all z > 0 and i = 1, 2 . . . . . The derivation of (8) has some similarities to the derivation of Lemma of Block et al. [5]. For the infinite-horizon case we want to find an n which minimizes B(n; p), the total expected long-run cost per unit time. Recall that n-2 C2+

B(H;

E

i=O

p) =

fO hi+l(z)PP(z)Pi(z)q(z)r(z) n-1

dz (9)

oo

z fo

i=O

We see that the inequalities

L(n; p) >__c2 and

B(n + 1; p) > B(n; p)

L(n-1;

and

B(n; p) < B(n -

1; p) if and only if

(lo)

p)
where foOO

n--I

oo

hn(Z)Fp(Z)Pn_l(z)q(z)r(z ) dz E fo FP(Z)Pi(Z) dz

L(n; p) =

fo Pp( z)p.( z) dz |

(11)

n-2

[ 0 -i~=O~ h i + l ( Z ) F P ( Z ) P i ( z ) q ( z ) r ( z ) d z

for 1 , 2 ,nf =o=r O.

"'"

Further,

L(n+

1;

n

p) -L(n; p) ~

[ foh.+l(Z)Fp(z)pn(Z)q(z)r(z) dz

= z fo p (z)pi(z)dzl i=0

fo FP(Z)Pn+I( Z) dz fo~h.(z)ffp(z)pn_l(z)q(z)r(z) dz (12)

JOPp(Z)pn(Z) dz We also require the following result which is useful for proving the Theorem.

S.H.Sheu / Determiningoptimalnumberof minimalrepairs

42

Lemma 1. If h~(z) is continuous and differentiable in z and hn(z)r(z)- h'n(Z) is increasing in z and n, where h'(z) = (dh~(z)/dz) then

fo°~hn(z)ffp(Z)Pn_l(z)q(z)r(z) dz fo°~ffp(z ) p.( z ) dz is &creasing fl~ n and lim

n---~¢~

fo°~hn(z)ffP(Z)Pn-x(Z)q(z)r(z) dz oo f ifo( z )P , ( z )

Jo--

= ho~(oo)r(oo)- h:(oo).

dz

Proof. The expression

fo°Chn(z)ffp(Z)Pn_l(z)q(z)r(z) dz fo~ffp(z)p.( z) dz can also be written in the useful alternative form [

z

~n-1

fo~hn(z) exp(_ f:r(x) dx) Ifoq(x)r(x) dx) -(-n=~)~. oo z foeXp(-for(X)

q(z)r(z) dz

(Soq(x)r(x) dx )n dx}

n!

(13)

dz

Integrating by parts, we have

oo z hn(z)exp{-fo r ( x ) d x }

fo

(SoZq(x)r(x)dx -~)!

q(z)r(z) dz

)n

= fo (hn(z)r(z)-hn(z))exp(- foZr(x) dx) (f°Zq(x)r(x) dx ni

dz

Thus, we only need to show that

fo°°(hn(z )r(z ) - h'n(z ) ) exp{ - f:r(x ) dx } ( ]:q(x )r(x ) dx ) n dz (14)

foeXp( - f:r( x ) dx } ( ]:q( x )r( x ) dx )"

dz

is increasing in n when the conditions of Lemma 1 are satisfied.

S.H.Sheu / Determiningoptimalnumberof minimalrepairs

43

Let

J(T) = {/0T( hn+l(Zlr(z ) -h'n+l(Z))

exp

{ S0zr(x) dx }(S0zq(x)r(x) -

dx )n+l dz

×SoTexp{ior(x)dx}{i;q(x)r(x )dx)"dz) -{foT(h.(z)r(z)-h'.(z)) exp{foZr(x)dx)(f~:q(x)r(x)dx)" dz ×fo'exp(-foZr(x) dx}(foZq(x)r(x) dx)'+' dz}.

(15)

T h e n it is easily seen that J(0) = 0 J ' ( T ) > 0. Thus, J ( T ) > 0 for all T > 0, and hence expression (14) is increasing in n. T h e r e f o r e we have that

fo h,(z)PptZ)p,_i(z)q(z)r(z)

dz

dz is increasing in n. Next, we show that

fo~h,( z)ffp( Z)p,_l( z)q( z)r( z) dz lim

fo FP(z)ffn(Z) dz ~(h~(z)r(z)-h'n(z)) lim

j; exp {jo

exp{-~r(x)dx}(foZq(x)r(x)dx)"

r(x) dx }(j0zq(x)r(x) dx

dz

) dz (16)

= h~(~)r(~) - h'(~).

Evidently,

f;( hn( z ) r( z ) - h'~(z ) ) exp{ -

flr(x)dx}(foZq(x)r(x)dx)" dz
fo=eXp(--for(X) dx}(foZq(x)r(x)dx) n dz for n = 1, 2, . . . . On the other hand, for any T ~ (0, oo), we have

f;(h,(z)r(z)- h',(z))exp{- for(x) dx} ( foq(X)r(x) dx)n dz f~exp(- for(x)dx}(foq(X)r(x)dx)"dz = [foT(h.(z)r.(z)-h'(z))exp(- for(x)dx} ( foq(X)r(x) dx )" dz

(17)

44

S.H. Sheu / Determiningoptimalnumberof minimalrepairs

+j;(hn(z)r(z)-hrn(Z))exp(-SoZr(x)dx)(Soq(X)r(x)dx)ndz] >([SoTxp(-for
n

n

f0Texp(- f0zr(x )d x }(f0q(x)r(x)dx ) dz+ f; exp( - fo~r(x )d x )(fo zq(x)r(x)dx ) dz hn(T)r(T) - h'.(T)

(18)

1+ f°Texp(-f°r(x'dx)(j°q(x)r(x)dx)ndz

I

f;exp(-foZr(x)dx)(joq(X)r(x)dx)"dz

I"

Further, the bracket of the denominator is, for T < T1,

z

Z

foreXp(-for(X) dx)(foq(X)r(x ) dx /;exp(-foZr(x) dx <

)n

dz

)(SoZq(x)r(x)dx )ndz

foTexp(--foZr(x) dx)(foZq(x)r(x)dx)

~ dz

f=exp(-for(X ) dx)(foq(X)r(x ) dx) n dz TI

<

t

ioTexp(--f:r(x) dx} dz }n

foriq(x)r(x)dx forq(x)r(x)-~x

~

~

--+ 0

as n ~ .

(19)

dx} dz

fr'exp(-f°r(x)

Thus, from (17), (18), and (19), we have h=(~) r(oo) - h'(oo)

n ---~oo(hn(oO)r(oo) - h'(oo))

= lim

j;(h,( z)r( z) - h'n(z ) ) e x p ( - f o r ( X ) d x ) ( j o q ( X ) r ( x >__

lim

n-+oo

f0~e , ~

>__h~( r ) r ( r)

(S;r)(So z -

(x) dx

q(xlr(xldx

) dx ) n dz

)n

dz

- h'(r),

which implies (16) since T is arbitrary. Remark 2. If satisfied.

h,(z)

is constant and

r(z)

is monotone increasing, then the conditions of Lemma 1 are

45

S.H. Sheu / Determining optimal number of minimal repairs

We seek the optimal n u m b e r n*(p) which minimizes B(n; p) in (9) under certain conditions. To

f0 pp(z)dz. oo

simplify the equations, we write/Zp =

Theorem 1. Suppose that hn( z ) is continuous and differentiable & z and hn( z )r( z ) - h'n(z ) is increasing in z and n, where h'n(z) = (dhn(z)/dz). Then if h3~) r(~) > - -

1

h~(o~)

[

+ h®(~)/./,~ C2+ i=)

ooh

i+l(z)ffp(z)pi(z)q(z)r(z)

]

dz ,

there exists a finite and unique n*( p) which satisfies L ( n ; p ) > _ c 2 and L ( n - l : p ) < c 2 ,

n=l,2,

(20)

...

Proof. The inequalities B(n + 1; p) > B(n; p) and B(n; p) < B(n - 1; p) imply (20). Further, if the conditions of the theorem are satisfied, then from expression (12) and L e m m a 1, it is easily seen that L(n; p) is an increasing function of n and lim L ( n ; p) =(ho~(oo)r(oo) -h:(oo))Ixpn---~

~ofo h i + , ( z ) f f p ( Z ) p i ( z ) q ( z ) r ( z ) dz. i=

Thus, if

[

h 3 ~) 1 r ( ~ ) >. - h~(o~) + h~(°°)tX--p c2+ i=

~h

i+l(Z)Fp(z)pi(z)q(z)r(z)

]

dz ,

then a solution to (2) exists and, from the monotonicity of L(n; p), it is unique. R e m a r k 3. We have assumed that the replacement costs for both cases of the n-th type 1 failure and type 2 failure are the same. In reality, they may be different from each other. We suppose that c 2 is the replacement cost of the n-th type 1 failure and c 3 is the replacement cost of the type 2 failure. Then, the total expected cost per unit time in (9) is rewritten as B(n;p)=

c2

ffp(Z)pn_l(z)q(z)r(z)dz+c

+ ~-~ fo h i + l ( z ) f f P ( z ) P i ( z ) q ( z ) r ( z ) i=O

[J:

3 1-

ffp(Z)Pn_l(z)q(z)r(z)dz

dz

]

z ) p i ( z ) dz. i

(21) We can discuss the optimal numbers which minimize B(n; p) by a similar method, although we omit this here.

3. Special cases

Case 1: p ( z ) = 0 and g(C(z), ci(z))= c I. Makabe and Morimura [6,7,8] considered this case in which only minimal repairs with fixed cost c 1 are p e r f o r m e d on failure before the n-th failure, whereas a unit is

S.H. Sheu / Determining optimal number of minimal repairs

46

replaced at the n-th failure. In this case, if we put p ( z ) = 0 and hi(z)---c! in (9), then we get the following result, which Makabe and Morimura obtained:

B(n; p )

c 2 + (n - 1)c 1 =

(J0

n--1 ~ -E [ i=0

i

r(x) dx

)(J0z )

n = 1, 2 . . . .

(22)

r(x) dx

dz

i!

~0

,

Park [12] also considered this case when F(x) is a Weibull failure distribution.

Case 2: p(z) =p and g(C(z), Ci(Z)) = C1. Nakagawa [10] considered this case in which type 1 occurs with probability q = 1 - p and is removed by minimal repair at a cost c 1, and type 2 failure occurs with probability p and is removed by replacement. The unit is replaced at the times of type 2 failure or n-th type 1 failure, whichever occurs first. In this case, if we put p(z) =p, q(z) =q = 1 - p , and hi(z) = c 1 in (9), then we get the following result (Nakagawa [10]): c 2 + c , [ ( q - q n ) / ( 1 - q)] B(n; p )

i

=

~lqif i=0

,

n = 1, 2.

(23)

exp(-foZr(x) d x ) ( f : r ( x ) d x ) 0

i!

dz

Case 3: p ( z ) = p and g(C(z), Ci(Z))= C. This is the case considered by Park [13]. H e r e we discuss a model where, at failure, one replaces the unit or repairs it depending on the random cost C* of repair. Let c be the constant repair limit. A replacement at failures takes place if C* > c, and if C* < c, then one proceeds with minimal repairs. Suppose that the random repair cost C * has distribution K*(z), density function k * ( z ) and finite mean. Let 1 - p = K *(c) be the probability of minimal repair (i.e. type 1 failure) at failure. Then the r a n d o m repair cost C of minimal repair has density k(z)= k * ( z ) / ( 1 - p ) for O < z < c and hi(z)=E[C]. In this case, if we put p ( z ) = p = 1 - K * ( c ) , q ( z ) = q = 1 - p , and hi(z) = E[C] in (9), then we get the following result (Park [13]): C2 -[-E[C]q(1 - qn-1)/(1 - q) i

=

B(n;p)

n-~

,

n=l,2

....

(24)

ooexp( - foZr(x) dx}(foZr(x) dx )

E qifo

i!

~

dz

i=0

Case 4: g(C(z), ci(z)) = C(z) + ci(z). In this case, if we put hi(z) = E[C(z)] + ci(z) in (9), then we get the following result:

n--2 C2+

B(n; p )

E

£~°(E[C(z)]

+Ci+l(Z))ffp(z)Pi(Z)q(z)r(z) dz

i=0 =

n-1

E£

i=O

ot~

(25)

dz

4. A numerical example In the numerical analysis here we shall consider the system with Weibull distribution which is one of the most common in reliability studies and compare the results with those obtained in Nakagawa [10].

47

S.H. Sheu / Determining optimal number of minimal repairs

The p.d.f, of the Weibull distribution with shape parameter/3 and scale parameter 0 is given by

f(t)=-~

(ot i (ot1 exp -

,

t>0,

(26)

/3,0>0,

and the parameter of the distribution will be chosen to b e / 3 = 2. Suppose that g(C(z), ci(z)) = C(z) + ci(z). Here we discuss a model where, at failure, one replaces the system or repairs it depending on the random cost C of repair. Let c= be the constant cost. A replacement (type 2 failure) upon failure at age z takes place if C > 6(z)cG if C < 6(z)c=, then one proceeds with a minimal repair (type 1 failure). The parameter 3(z) can be interpreted as a fraction of the constant cost c= at age z, and 0 < 6(z) < 1. Here we consider the following parametric form of the repair cost limit function 3(z) = ~e-"Z with 0 < 6 < 1 and a > 0. Suppose that the random repair cost C has a normal distribution L(.) and density function l(.), with mean /z and standard deviation o- (the probability of a negative cost is negligible). If an operating system fails at age z, it is either replaced with a new system with probability p(z)

= 1 - ' of'~(z)c='" l(x) dx,

(27)

or it undergoes minimal repair with probability q(z)

= f~(z)c='" t( x) dx.

(28)

Then the random part C ( z ) of minimal repair cost at age z has density

t(x) t:(x)

-

q(z)

for

and

hi(z ) = E [ g ( C ( z ) , ci(z)) ] = f~(z)C=xlz(x) d x + c i ( z ) "0

1 f~(z)c ~xt(x) ..

q(z)

d x + c i ( z ).

(29)

JO

It is noteworthy that when a ~ 0 and ci(z) = 0, we have 6(z) = 3, so that this model reduces to special case 3 considered by Park [13]. Furthermore, when a --* 0, coo--~ ~ and 6 = 1, it reduces to special case 1 considered by Makabe and Morimura [6,7,8]. It can be seen that this model is an improvement on previously known policies. Finally, we compute the optimal number n*(p)which minimizes the expected cost B(n; p) in (21). In our computations we consider the following two different cases: Case a. c 2 = 1000, c= = 1000, c 3 = 1000, 1200, 1500, C - N ( 1 0 0 , 202), ci(z) = 0, 0 = 1; Case b. c 2 1000, c= = 1000, c 3 1200, C ~N(200, 402), C i ( Z ) = O , C i ( Z ) = 2z, Ci(Z)= 10i + 2z, 0 = 1012.2. The parameters 8 and a were varied to take different values in order to see their influence on the optimal solution. The results are given in Tables 1 and 2. It should be noted that there is a slight difference between the n*(p)-values obtained in Nakagawa [10] and those of our results in Table 1 for the case a = 0. In the present study, the cost of minimal repair is random and the expected cost of minimal repair is varied for the different probability q of type 1 failure. Nakagawa considered that the cost of minimal repair is constant and the same for the different probability q of type 1 failure. In this paper, the repair cost limit function 3 ( z ) = ~e T M is chosen for the purpose of easy computation. From the numerical results, we can derive the following remarks: (i) Some improvement can be made in the minimum cost per unit time if one allows for minimal repair at failure; =

=

S.H. Sheu / Determining optimal number of minimal repairs

48

Table 1 Optimal number of minimal repair before replacement with general random repair cost, for the Weibull distribution, c 2 = 1000, c= = 1000, c 3 = 1000, 1200, 1500, C ~N(100, 202), ci(z) = O, 0 = 1 Nakagawa's results (Constant repair cost) q

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

Our results (General random repair cost)

¢3 = 1000

c 3 = 1200

c 3 = 1500

n*(p)

n*(p)

n*(p)

10 11 13 15 17 20 22 24

10 9 8 8 7 7 7 6

10 7 6 5 4 4 3 3

3

a

1 0.12563 0.11683 0.11049 0.10507 0.1 0.09493 0.08951

0 0 0 0 0 0 0 0

c 3 = 1000

c a = 1200

c 3 = 1500

n*(p)

B(n*,p)

n*(p)

B(n*,p)

n*(p)

10 11 14 17 18 20 20 20

608.7121 643.2690 691.7741 748.6493 807.8384 866.0926 922.4286 976.7068

10 9 9 9 8 8 8 7

608.7123 10 698.5302 7 786.7158 6 870.7396 5 950.3322 4 1025.7933 4 1097.4846 3 1165.8213 3

B(n*,p) 608.7126 775.5013 917.7276 1042.9190 1156.1650 1260.2251 1357.1947 1452.5131

Table 2 c 2 = 1000, c= = 1000, C3 = 1200, C~N(200, 402), 0 = 1012.2

6

a

1 1 0.25126 0.25126 0.2336 0.23366 0.22098 0.2209 0.21013 0.21013 0.2 0.2 0.18986 0.18986 0.17902 0.17902

0 0.00015 0 0.00015 0 0.00015 0 0.00015 0 0.00015 0 0.00015 0 0.00015 0 0.00015

Ci(Z) = 2Z

ci(z) = 0

Ci(Z) = I0i +2Z

n*(p)

B(n*, p)

n*(p)

B(n*, p)

n*(p)

B(n*, p)

5

0.8159

5

0.8656

4

0.9181

4

0.9711

4

1.0242

4

1.0770

4

1.1297

4

1.1820

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2.2274 2.2274 2.1746 1.9248 2.1172 1.8137 2.0534 1.7278 1.9823 1.6545 1.9028 1.5893 1.8139 1.5299 1.7144 1.4755

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2.2348 2.2348 2.1815 1.9304 2.1235 1.8184 2.0592 1.7318 1.9874 1.6578 1.9073 1.5920 1.8176 1.5320 1.7173 1.4755

(ii) f r o m T a b l e 1, t h e m i n i m u m c o s t p e r u n i t t i m e will b e r e d u c e d w h e n t h e p r o b a b i l i t y o f m i n i m a l r e p a i r i n g is g r e a t e r ; (iii) f r o m T a b l e 2, t h e m i n i m u m c o s t p e r u n i t t i m e will b e r e d u c e d w h e n t h e p r o b a b i l i t y o f m i n i m a l r e p a i r i n g is a g e - d e p e n d e n t ; (iv) it c a n b e s e e n t h a t t h e p r e s e n t m o d e l s a r e a g e n e r a l i z a t i o n o n p r e v i o u s l y k n o w n p o l i c i e s .

Acknowledgements The author thanks the referees for their valuable comments, which greatly enhanced the clarity of the article. This research was supported by the National Science Council of the Republic of China (NSC 80-0415-E-011-15).

S.H. Sheu / Determining optimal number of minimal repairs

49

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