ELSEVIER

An International Joumal Available online at www.sciencedirectcom computers &

. r Cd) o , .=c . , mathemat ics with applications

Computers and Mathematics with Applications 51 (2006) 345-356 www.elsevier.com/locate/camwa

Statist ical Est imat ion Algor i thms for Repairs -Time Limit

Rep lacement Scheduling under Earning Rate Criteria

T A D A S H I D O H I AND A K I R A A S H I O K A Depar tment of Information Engineering

Graduate School of Engineering Hiroshima University, 1-4-1 Kagamiyama

Higashi-Hiroshima 739-8527, Japan dohi�9 hiroshima-u, ac. jp

NAOTO KAIO Depar tment of Economic Informatics

Faculty of Economic Sciences Hiroshima Shudo University

1-1-1 Ozukahigashi, Asaminami-ku Hiroshima 731-3195, Japan

kaio�9 ac. jp

S H U N J I O S A K I Depar tment of Information and Telecommunication Engineering Faculty of Mathemat ical Sciences and Information Engineering

Nanzan University, 27 Seirei-Cho Seto 489-0863, Japan

shunj iOit. nanzan-u, ac. j p

A b s t r a c t - - I n this paper, we formulate two repair-time limit replacement problems with imperfect repair under earning rate criteria with and without discounting. First, the optimal repair-time limits which maximize the long-run average profit rate and the expected total discounted profit over an infinite time horizon are analytically derived. Next, we develop the nonparametric algorithms for estimating the optimal repair-time limits, provided that the complete sample data of repair time are given. The basic idea is to apply the Lorenz statistics and to transform the underlying algebraic problems to the graphical ones. Finally, we present some numerical examples to show that the proposed algorithms can be useful to estimate the profit-based repair limit replacement schedule. @ 2006 Elsevier Ltd. All rights reserved.

K e y w o r d s - - M a i n t e n a n c e , Repair-time limit, Imperfect repair, Earning rate, Lorenz statistics, Nonparametric.

This research was partially supported by the Ministry Education, Culture, Sports, Science and Technology, Grant- in-Aid for Exploratory Research; Grant No. 15651076 (2003-2005), Scientific Research (B); Grant No~ 16310116 (2004-2006), Scientific Research (C); Grant No. 16510128 (2004-2005), Nanzan University Pache Research Subsidy for 2005, and Research Program 2005 under the Institute for Advanced Studies of Hiroshima Shudo University.

0898-1221/06/$ - see front matter @ 2006 Elsevier Ltd. All rights reserved. doi: 10. 1016/j.camwa.2005.11.004

Typeset by .43,~-TEX

346 T. DOHI et al.

1. I N T R O D U C T I O N

Since the seminal contribution by Hastings [1], a large number of repair limit replacement prob- lems were considered in literature [2-6]. Nakagawa [2] considered a simple repair-time limit replacement problem under an earning rate criterion. Nguyen and Murthy [4] analyzed a differ- ent repair-time limit policy with imperfect repair. In this paper, we focus on a mixed model of Nakagawa [2] and Nguyen and Murthy [4]. Consider a single-unit system where each spare unit is provided only by an order after a lead time, and each failed unit is repairable. When the unit fails, the decision maker (DM) estimates the completion time distribution of repair, which may be a possibly subjective one. If DM estimates that the repair is completed up to a prespecified time-limit, the repair is started immediately, otherwise, the spare unit is ordered with a lead time. Since the repair is imperfect, the unit repaired or even ordered can fail again during a finite time horizon. The problem for DM is to determine the optimal repair-time limit which maximizes any earning rate criterion.

Dohi et al. [7,8] considered the above models with subjective repair time distribution under the expected cost criteria, and developed estimators of the optimal repair-time limits, by applying the Lorenz statistics or the Lorenz curve. Since the knowledge on the repair-time distribution is incomplete in general, such a statistical estimation method for the optimal repair-time limit will bc useful in the practical maintenance situation [9]. The Lorenz curve was first introduced by Lorenz [10] to describe income distributions. Since the Lorenz curve is essentially equivalent to the Pareto curve used in the quality control, it will be one of the most important statistics applied in every social sciences. The more general and tractable definition of the Lorenz curve was made by Gastwirth [11]. Goldie [12] proved the strong consistency of the empirical Lorenz curve and discovered its several convergence properties. Chandra and Singpurwalla [13] investigated the relationship between the total time on test statistics [9] and the Lorenz statistics, and derived a few aging and partial ordering properties.

In this paper, we formulate the repair-time limit replacement model with imperfect repair [8] under earning rate criteria with and without discounting. First, after describing the notation and assumptions used here, the optimal repair-time limits which maximize the long-run average profit rate and the expected total discounted profit over an infinite time horizon are analytically derived. Next, we develop the nonparametric algorithms for estimating the optimal repair-time limits, provided that the complete sample data of repair time are given. The basic idea is to apply the Lorenz statistics and to transform the underlying algebraic problems to the graphical ones. Finally, we present some numerical examples to show that the proposed algorithms can bc useful to estimate the profit-based repair limit replacement schedule.

2. R E P A I R - T I M E L I M I T R E P L A C E M E N T M O D E L

2.1. N o t a t i o n

The repair time X for each unit is a nonnegative i.i.d, random variable. The decision maker (DM) has a subjective probability distribution function Pr{X < t} = G(t) on the repair time, with density g(t) (> 0) and finite mean 1/~ (> 0). Suppose that the distribution function G(t) E (0, 1) is arbitrary, continuous and strictly increasing in t E (0, c~), and, in addition, has an inverse function G- l ( . ) . Further, we define

to E [0, oo): repair-time limit (decision variable), Fl(t), f l( t) , 1/pl (> 0): c.d.f., p.d.f., and mean of time to failure for a repaired unit, F2(t), f2(t), 1/p2 (> 0): c.d.f., p.d.f., and mean of time to failure for a new (spare) unit,

k (> 0): penalty cost per unit time when the system is in down state, e0 (> 0): earning rate per unit operation time, el (> 0): repair cost per unit time, c (> 0): fixed cost associated with the ordering of a new unit,

Statistical Estimation Algorithms

" " ~ ~ , t i m e X l ip: q-. -q

347

. . . . . . . . i . . . . . . . . . 1/~; time

X : failure (renewal point)

�9 : recovery point for a unit - - : operation period m : repair period

�9 -- : lead time

Figure 1. Configuration of repair-time limit replacement with imperfect repair.

L (> 0): lead t ime for delivery of a new unit,

(> 0): discount rate, /2{r = f o e x p ( - ~ t ) ' O ( t ) d t for an a rb i t r a ry continuous function y)(.)

(Laplace t ransform of ~( . ) ) ,

~(-): = 1 - r (survivor function).

2.2. M o d e l D e s c r i p t i o n

Consider a single-unit repairable system, where each spare is provided only by an order after a lead t ime L and each failed unit is repairable. When the unit has failed at t ime t = 0, the DM

wishes to de termine whether he or she should repair it or order a new spare. If DM est imates tha t the repair is completed within a prespecified t ime limit to E [0, co), then the repair is s ta r ted immedia te ly at t = 0 and completes at t ime t = X. After the complet ion of repair , the unit is s ta r ted to opera te again, bu t can fail again for a finite t ime span since the repair is imperfect.

Then, the mean failure t ime is l / p 1 .

On the other hand, if DM es t imates t ha t the repair t ime exceeds the t ime l imit to, then ti le failed unit is scrapped at t ime t = 0 and a new spare unit is ordered immediately. A new unit is delivered after the lead t ime L. Fur ther , the new unit can also fail for a finite t ime span and then the mean failure t ime is 1/#2. Wi thou t any loss of generality, it is assumed tha t the t ime required

for replacement of a failed unit can be negligible. Under these model set t ing, we define the interval from the failure t ime to the following failure t ime as one cycle. Figure 1 depicts the configuration of the repa i r - t ime limit replacement problem with imperfect repair under considerat ion.

3. A N A L Y S I S

3.1. L o n g - R u n A v e r a g e P r o f i t

Define the t ime interval from a failure point to the next failure point as one cycle. Suppose

tha t the same cycle is repeated again and again over an infinite t ime horizon. Since the failure

point can be regarded as a renewal point, the mean cycle length is given by

jo to io Io TL (to) = t dG(t) + G( to )F l ( t ) dt + L dG(t) + G(to)fi'2(t) dt

= t dG(t) + a ( t o ) + ( L + )Q(to). (1)

348 T. Dora et al.

Also, the expected total profit during one cycle becomes

{I7 /7 } io '~ VL(to) = eo G(to)Fl(t) dt + G(to)/~2(t) dt - (el + k). t dG(t)

f7 I7 - k. L dO(t) - c dO(t) (2)

= ~-TG(t~ + { e~ - (c + kL) } O(t~ - (e~ + k) fo t~

From the familiar renewal reward argument, the long-run average profit rate (the expected total profit per unit time in the steady state) is given by

TPL(t0) = l i m t ~ E[total profit on (0, t]] = VL(to)/TL(tO), (3) t

and the problem is to derive the optimal repair-time limit t~ E [0, co) satisfying

TPL(t~) = max TPL(t0). (4) 0 < t o < ~

Differentiating TPL(to) with respect to to yields dTPL(to)/dto = g(to)qL(tO)/TL(to) 2, where

qL(tO) = c+ kL + eo ~ - (el + k)to TL(to)

- ( t ~ lpl- #2-1) VL(t~

(5)

Then, we have the following result on the optimal repair-time limit under the assumption VL(to) > 0 for all to.

THEOREM 1. There exists a finite and unique optimal repair-time limit t~ (0 < t~ < z~) satisfying qL ( t~) = O, and the corresponding maximum long-run profit rate is given by

TPL(t~) = eo (1/ttl -- 1/#2) + c + kL - (el + k)t~ t~ - L + 1/pl - 1/#2

(G)

3.2. Tota l D i scoun t ed Pro f i t

Next, consider the case where the profit is discounted with the discount rate/3 (> 0) over an infinite time horizon. Then, the expected total discounted profit during one cycle is given by

'

VD(tO) = e0 exp{-~(x + y)} dy dE1 (t ) dG(x)

f;io /o ' + e0 exp{-~(L + y)} dy dF2(t) dG(x) - cexp{-~L} dG(t)

L'o/o ' f ; /o - (el + k ) e x p { - f l x } d z d G ( t ) - k e x p { - Z x } d x d G ( t )

= e_o~(1 - z;{A(~)}) exp{-~x} dG(x) - (1 - exp{-~t}) da(t)

eo exp{-~L} (1 - s + r

_ [k(1- exp{-~L})~ + cexp{-gL}] e(to)

(7)

Statistical Estimation Algorithms 349

Since the expected present value of the unit profit just after one cycle is given by

5(to) = exp{-3(t + x)} da(t) dF1 (z) J0 J0

+ exp{-3(L + z)} da(t) dY2(x)

fo '~ = s exp{-3 t} de( t ) + C{/~(3)} exp{-3L}G( to) ,

the expected total discounted profit over an infinite time horizon becomes

vo(to) TPD(to) = E VD(tO)6(to)J

j=o ~(to)

It is evident that TPL(t0) = l i ~ 3" TPD(t0).

Of our interest is the derivation of the optimal repair-time limit t; E [0, oc) satisfying

(s)

(9)

(10)

TPD(t~)) = max TPD(to). (11) 0<to<OC

Similar to equation (5), it is seen that dTPD(to)/dto = g(to)qD(to)/5(to) 2, where

qD(tO) = { ~ ( 1 -- s exp{--3to} eo exp{-3L} (1 - s 3

+ c e x p { - 3 L } - k +/3 e-----'-~l (1 - exp{-3t0}) + k(1 - exp{-3L})3 } 5(to) (12)

- ( s exp{-3L} - s (3)} exp{--3to})VD(tO).

THEOREM 2. Suppose that VD(to) > 0 for all to.

(i) I f qD(O ) > 0 and qD(O0) < O, then there exists a finite and unique optimal repair-time limit t~ (0 < t~ < oc) satisfying qD(t~) = O, and the corresponding expected total discounted profit over an infinite time horizon is given by

TPD(t ; ) = [Co(1 - s exp{-3 t ;} - co(1 - s exp{ -3n}

- ( k + el)(1 - exp{-3 t ;} ) + k(1 - exp{-3L}) + c3exp{-3L}] (13)

/[3(s exp{-3L} - s (3)} ex p { -3 t ; } ) ] .

(ii) L~ qD(O) <_ O, then the optimal repair-time limit is t~ = 0 and TPD(t~) -- TPD(0). (iii) If qv(OC) >_ O, then the optimal repair-time limit is t~ --+ oo and TPD(t~) = TPD(OC).

In this section, we derived the optimal repair-time limit replacement policies maximizing two kinds of profit functions. It should be noted that the repair time distribution has to be completely known, if one calculates the optimal policies according to Theorem 1 and Theorem 2. However, in general, it is not so easy to identify the repair time distribution for a highly reliable equipment. In the following section, we will develop nonparametric algorithms to estimate the optimal repair- time limits under respective profit functions.

4. G R A P H I C A L M E T H O D S

For a continuous repair time c.d.f., p = G(to), define the Lorenz transform,

= A f o G-1 (p) r t dG(t), (14)

350 T. DOHI et al.

where G-l(p) = i n f { t _ > 0 : G(t) > p}, (15)

if the inverse function exists, and t /A = fo~ Then, the curve s = (p,r E [0, 1] • [0,1] is called the Lorenz curve. From a few algebraic manipulat ions, we obtain the following useful result to interpret the underlying optimizat ion problem max0<~o<ooTPr( t0) geometrically.

THEOREM 3. If (1/#1 -- 1/#2)(e0 + el + k) + c - elL > O, then obtaining the optimal repair- thne limit to* maximizing the long-run average profit ra te TPL(to) is equivalent to obtaining p" (0 < p* _< 1) such as

min r +/hL, (16) 0 < p < 1 p -1- OL L

otherwise,

wh ere

max r + / 3 L , (17) 0_<p_<l p 3c O~ L

~L = - 1 + (eo + el + k) /# l

J~L =

( 1 / p l - 1 / # 2 ) (eo + el + k) + c - e l L '

(A/pl){(eo + k)L + c} ( 1 / # 1 - 1 / # 2 ) (e0 + e l + k) + e - e lL"

(18)

(19)

Theorem 3 is the dual of Theorem 1. From this result, it can be seen tha t the opt imal repair- t ime limit to* = G-I(p *) is determined by calculating the opt imal point p* (0 < p* _< 1) minimizing or maximizing the tangent slope from the point (--aL, --/hL) to the curve (p, r C [0, 1] x [0, 1].

For the discounted case, define the modified Lorenz transform,

fo c- l (v) exp{- /5 t } dG(t)

r =-- f : - , ( z ) exp{-3t}dG(t )" (20)

THEOREM 4. If (eo + el + k) exp{-/hL}L{f2(/5)} - [el + (eo + k - c/5) exp{-~L}]f -{ f l (~)} > O, then obtaining the optimal repair- t /me limit to* maximizing the expected total discounted profit over an infinite time horizon TPD(t0) is equivalent to obtaining" p* (0 <_ p* <_ 1) such as

m a x : o<_v<_l p + C~D

(21)

otherwise

w h e r e

min : r + ~D (22) 0_<p_<l P q- ~ D

aD = (s -- 1)(k + eo + e l ) / ( [ e i + (el + k - c/?)exp{-/hL}] s

-(eo + el + k) exp{-/hL}g{f2(/5)}) - 1, (2a)

/5D = [(e0 + el + k) exp{-/hL}E{f2 (/5)} - (k + e0 - c/5) exp{- /hL} - el]

/[s (/5)} ([el + (eo + k - c/5)exp{-/hL}] Z:{fl (/5)}

- (eo + el + k) e x p { - ~ L } s (/5)})1-

(24)

Sta t i s t ica l Es t ima t ion Algo r i t hms 351

Next, suppose that the optimal repair-time limit has to be estimated from n ordered complete observations: 0 = x0 <_ xl _< x2 _< " - <_ xk of the repair times from a continuous c.d.f. G, which is unknown. Then, the empirical distribution for this sample, is given by

- , for xi < x < Xi+l, (25) a ~ ( z ) - n

1, for x~ <_ x,

where i = 0, 1, 2 , . . . , n - 1. Then, the Lorenz statistics [12,13] can be defined by

r = E xi xi , (26) i=i

where [a] denotes the least integer not smaller than a (ceiling function). From equations (25) and (26), plotting ( i /n , r (i = 0, 1, 2 . . . . , n) on 7~ 2 and connecting them by line segment yield the sample Lorenz curve Z:~ = ( i /n , r E [0, 1] x [0, 1]. From the analogy of Theorem 3, we obtain an estimator of the optimal repair-time limit which maximizes the hmg-run average profit rate as follows.

THEOREM 5. Define the estimator of the optimal repair-time limit by [~ = x~. I f (1/#1 - 1/#2)(eo + el + k) + c - e lL > O,

otherwise,

{x$l min r + ~L } (27) 0<~<n i / n + oi L '

r + ZL } (28) i / n + aL '

the sample mean,

x r l m a x - - 0<i<n

where 1/A ill flL (equation (19)) is replaced by

i=1

In fact, Goldie [12] proves the strong consistency of the Lorenz statistics given in equation (26), that is, ~in --* r as n --* oc, a.s. This fact means that the estimator of the optimal repair- time limit t ; may be also consistent. The graphical procedure proposed here has an educational value for better understanding of the optimization problem and it is convenient for performing sensitivity analysis of the optimal repair-time limit when different values are assigned to the model parameters. The special interest is, of course, to estimate the optimal repair-time limit without specifying the repair time distribution. Although some typical theoretical distribution functions such as the lognormal distribution are often assumed for the repair time distribution, our nonparametric estimation algorithm can estimate the optimal repair-time limit based on the on-line knowledge about the observed repair times.

Next, define the statistics of the modified Lorenz transform by

( 2 9 )

/I[--~t (1-- J-- 1) (xJ--XJ-1)exp{--~32J} 1 n

By plotting ( i /n,r and connecting them by line segment, we obtain the modified sample Lorenz curve.

352 T. DOHI et al.

THEOREM 6. Define the es t imator o f the opt imal repair- t ime l imit by {~ = x*. I f

(eo + el + k ) e x p { - / 3 L } f~ {f2 (/3)} - [el + (Co + k - c/3)exp { - /3L}] f. { f l (/3)} > 0,

then

0<i<n i / n + O~ D '

{x '~[ m i n (]~i~n-[-/3D } (31) 0 < i < ~ i / n + a D "

In this section, we developed e s t i m a t i o n a lgor i thms for the op t ima l r epa i r - t ime l imits under

different e a rn in g ra te cri teria. As men t ioned before, it can be easily u n d e r s t o o d t h a t the es t ima-

tor t~ of the op t ima l repa i r - t ime l imit max imiz ing the long- run average profit ra te has s t rongly

consis tent , i.e., t~ --* t~ as n ~ oc. However, it is an open ques t ion whe the r the e s t ima te in

the d i scounted case can converge to the real o p t i m u m as n --~ oo. For the pract ica l purpose, we

will examine the convergence proper t ies of n o n p a r a m e t r i c e s t ima tors for the op t ima l repa i r - t ime

l imits t h rough a s imu la t ion s tudy in the following section.

5. I L L U S T R A T I V E E X A M P L E S

Suppose t h a t the repair t ime d i s t r i bu t ion is the following Weibul l d i s t r ibu t ion ,

c ( t ) = 1 - e - ( ~ / ~ (32)

where the scale p a r a m e t e r 0 = 1.2 a n d the shape pa r am e te r m = 2.2, The o ther pa ramete r s

are 1 / p l = 0.4300, 1 /p2 = 0.4500, e0 = 8.0000, el = 0.4000, k = 0.8000, c = 2.5000, and

L = 0.2500. We genera ted 20 sample from the Weibul l d i s t r i bu t ion above and regard t hem as

the repair t ime data . F igure 2a i l lus t ra tes an e s t ima t ion of the op t ima l repa i r - t ime l imit when

(eo + e l + k ) exp{- /3L}/ :{ f2( /3 )} - [ e , + (eo + k - e/3) exp{-f~L}] Z:{f , (~)} > 0. In th is case, since ( - -aL,- - /3L) = ( - -0 .7852,- -0 .8617) , it is observed t h a t {~ = x~ = 0.9382 and TPL( t~ ) = 2.1165.

Oil the o ther hand , F igure 2b depic ts an example for the case of (eo + el + k) exp{- t3L}Z:{f2 (~)} - [el + (eo + k - c/3) exp{- /3L}] s _< 0. The model pa rame te r s are O = 2.2, m = 2.0,

1 / # t = 0.3500, 1/i~2 = 0.7500, eo = 3.5000, el = 0.4500, k = 0.2000, c = 1.5000, and L = 3.0000. ^ .

For 20 sample from the Weibul l d i s t r ibu t ion , we get ( - a t , --/3L) = (1.9619, 1.6782), t~ = x 9 =

1.6052, and TPL( t~ ) = 0.1913.

Next, consider the d i scounted case. F igure 33 is an e s t ima t ion resul t for the case of (e0 + el +

k) e x p { - Z L } Z : { f 2 ( / ~ ) } - [el + (e0 + k - c / 3 ) e x p { - ~ L } ] L:{fl(/3)} > 0, where 0 = 15, m = 1.2,

/Z{f l (Z)} ---- 0.9550, ~2{12(/3)} = 0.9500, /3 = 0.0330, e0 = 6.5000, el = 0.3500, k = 0.3500, c = 9.8000, and L = 5.5000. In th is ease, we genera ted 50 r a n d o m n u m b e r s from G(t ) as the

repair t ime data . F rom ( - - a D , - - ~ D ) = (--0 .8828,--0 .7426) , we ob t a in t~) = X~s = 10.8862

and TPD({~) = 12.0033. In F igure 3b, we present an example for the case of (eo + el +

k ) e x p { - 3 L } s - [el + (eo + k - c 3 ) e x p { - ~ 3 L } ] s < 0, where 0 = 15, m = 1.2,

L:{fl( /))} = 0.9800, L:{f2(/3)} = 0.9750, /3 = 0.0500, eo = 5.0000, el = 0.5500, k = 0.1200, ^ .

c = 0.1250, and L = 10.0000. For 50 repair t ime d a t a from G(t) , we e s t ima te t~ = x 7 = 2.4629

and TPD( /~ ) = 1.0195 wi th ( - - a D , - - / 3 0 ) = (1.5027, 2.2648).

Final ly , we inves t iga te the convergence proper t ies of the es t imator , t~ = x~ for the d iscounted

case. In F igure 4, t he a sympto t i c behavior of the e s t ima te of op t ima l repa i r - t ime l imi t and its

associated expected to ta l d i scounted profit over an inf ini te t ime hor izon is i l lus t ra ted , where

O = 15, rrt -- 1.2, s --- 0.9550, L:{f2(/~)} = 0.9500, /3 = 0.0330, e0 = 6.5000, e, = 0.3500,

k = 0.3500, c = 9.8000, a n d L = 5.5000. In the figures, the hor izonta l l ines deno te the real

op t ima l r epa i r - t ime l imit and the cor responding m a x i m u m expected to ta l d i scounted profit over

an inf ini te t ime horizon. F rom these results, it is seen t h a t the op t ima l r epa i r - t ime l imit can be

e s t ima ted wi th higher accuracy when more t h a n 50 repair t ime d a t a are available.

Statistical Estimation Algorithms 353

1/~, = 1.0584 1/#1 = 0.4300 1//.t 2 = 0.4500 eo = 8.0000 el = 0.4000 k = O. 8000 c=2.5000 L = O. 2500 to =0.9382 C(to ) = 2.1165

~n

0.2587

-0.7852 0 0.4500 Pin

~........ . . . . . . . . . . . . . . . . . . . . . . . . . . . -0 .8617

(a)

~n

1.6782

0.2396

0 . . . . . . . . . . . . . i !

1/3. = 1.7403 1/#1 = 0.3500 1/#2 = 0.7500 eo = 3.5000 el = 0.4500 k = 0.2000 c = 1.5000 L = 3.0000 to = 1.6052

CGo) = O. 1913

0.4500 1.9619 Pin

(b)

Figure 2. Estimation of the optimal repair-time limit maximizing tile long-run aver- age profit.

354

L {g(fl) } = 0.6934 L{f~(fl)} = 0 . 9 5 5 0

L{.f2(fl)} = 0 . 9 5 0 0 fl = 0 . 0 3 3 0

L = 5 . 5 0 0 0 c = 9 . 8 0 0 0 k = 0 . 3 5 0 0

eo = 6 . 5 0 0 0 el = 0 . 3 5 0 0

to = 10 .8862

C(~o) = 12 .0033

-0.8828

T. D o r a et al.

0.5600 Pin

- 0 . 7 4 2 6

(a)

2 . 2 ~ 8

0 . 2 ~ 1

0

B

1

L { g ( f l ) ) = 0 . 5 8 4 5 L , f f l ( ~ ) = 0 . 9 8 0 0

LQ'2(f l)) = 0 . 9 7 5 0

/3 = 0 .05O0

L = 10 .0000 c = O. 1250

k = O. 1200 eo=5.0o00 el = 0 . 5 5 0 0

~o = 2 . 4 6 2 9

C(to) = 1 .0195

0.1400 1.5027 Pin

(b)

F igu re 3. E s t i m a t i o n of t he o p t i m a l r e p a i r - t i m e l imi t m a x i m i z i n g t he e x p e c t e d t o t a l

d i s c o u n t e d prof i t over an inf in i te t i m e horizon.

Statistical Estimation Algorithms 355

t~ = 11.840: \ A M . . . . . .

0 1 ~ 2 ~ 300 4(10 5 ~ 600 n

(a)

re~

100 200 300 400 500 600 n

(b)

Figure 4. Asymptotic behavior of the estimate of optimal repair-time limit and its associated expected total discounted profit.

R E F E R E N C E S

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