Optimal Spare Ordering Policy for Preventive Replacement Under Cost

8/16/2019 Optimal Spare Ordering Policy for Preventive Replacement Under Cost

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Optimal spare ordering policy for preventive replacement under costeffectiveness criterion

Yu-Hung Chien a, * , Jih-An Chen ba Department of Applied Statistics, National Taichung Institute of Technology, 129, Sec. 3, San-min Rd., North District, Taichung City 40401, Taiwanb Department of Business Administration, Kao-Yuan University, 1821, Chung-Shan Rd., Lu-Chu Hsiang, Kaohsiung County 821, Taiwan

a r t i c l e i n f o

Article history:Received 9 April 2008Received in revised form 3 June 2009Accepted 4 June 2009Available online 9 June 2009

Keywords:Ordering policySparePreventive replacementMinimal repairSalvageCost effectiveness

a b s t r a c t

This paper presents a spare ordering policy for preventive replacement with age-depen-dent minimal repair and salvage value consideration. The spare unit for replacement isavailable only by order and the lead-time for delivering the spare due to regular or expe-dited ordering follows general distributions. To analyze the ordering policy, the failure pro-cess is modelled by a non-homogeneous Poisson process. By introducing the costs due toordering, repairs, replacements and downtime, as well as the salvage value of an un-failedsystem, theexpected cost effectiveness in the long run arederived as a criterion of optimal-ity. It is shown, under certain conditions, there exists a nite and unique optimum orderingtime which maximizes the expected cost effectiveness. Finally, numerical examples aregiven for illustration.

2009 Elsevier Inc. All rights reserved.

1. Introduction

It is of great importance to avoid the failure of complex systems during the actual operation when such an event is costlyand/or dangerous. In such situations, one important area of interest in reliability theory is the study of various maintenancepolicies in order to reduce the operating cost and the risk of a catastrophic breakdown. Many preventive maintenance pol-icies have been proposed and discussed in the past four decades. See, Barlow and Hunter [1] , Barlow and Proschan [2] , Clé-roux et al. [3] , Beichelt [4] , Boland and Proschan [5] , Boland [6] , Nakagawa [7] , Nakagawa and Kowada [8] , Block et al. [9] , AitKadi and Cleroux [10] and Sheu et al. [11] , for example.

Most policies studied in the literature assume that at any time there is an unlimited supply of units available for replace-

ment; typically spare units are assumed to be inexpensive and bulk purchased under a stocking policy (Wang [12] ). How-ever, this might not be true on some occasions. For instance, when spare units are expensive and/or storage is costly, it isnatural in commercial industries that only one spare unit, which can be delivered by order, is available for replacement.In this case, we cannot neglect the random lead-time for delivering a spare unit. That is, it is essential and practical to intro-duce the random lead times. Once we take account of the random lead times, we should consider an ordering policy thatdetermines when to order a spare and when to replace the operating unit after it has begun operating.

Previous workson ordering policies have assumed that the costs for preventive and corrective replacements are equal (forexample, see Chien [13] , Dohi et al. [14] , Kaio and Osaki [15,16] , Kalpakam and Hameed [17] , Park and Park [18] , Thomas andOsaki [19] , Sheu [20] , and Sheu and Liou [21] ), which implies in essential that there is no particular need for preventive

0307-904X/$ - see front matter 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2009.06.017

* Corresponding author. Tel.: +886 4 22196660; fax: +886 4 22196331.E-mail address: [email protected] (Y.-H. Chien).

Applied Mathematical Modelling 34 (2010) 716–724

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replacement. Beside, most of them seek the optimum ordering policy by minimizing the expected cost per unit time in thelong run as a criterion of optimality. In general, however, the policy maximizing prot rate does not discriminate amonglarge and small investments. Thus, the policy that mentioned above might have this property. Therefore, the cost effective-ness as an alternative criterion is suitable for reecting efciency per dollar spent. The cost effectiveness is dened as

s availabilitys expected cost rate

(where ‘‘s-” implies statistical meaning) which reects the efciency per dollar outlay. Park and Park [18] and Chien andChen [22] also determined the optimal ordering time based on the cost effectiveness criterion. As described in Park and Park[18] , this criterion is useful for the effective use of available money. Especially, this criterion is useful when the benets ob-tained from investment are difcult to quantify. As an example, national security may benet from weapon systems (seeGrant et. al. [23] ), but expressing the gains in monetary terms is quite challenging.

In this paper, a general repairable system with general random repair cost is considered, and a spare ordering policy forpreventive replacement is presented for such a system. In the general repair model, when the system fails at age t , type Ifailure occurs with a probability of qðt Þ ¼ 1 pðt Þ and type II failure occurs with a probability of pðt Þ (0 6 pðt Þ 6 1), where pðt Þis a non-decreasing function in t . Type I failure is assumed to be minor, and can thus be rectied through minimal repair;while type II failure is catastrophic, and can only be removed by replacement. Minimal repair means that the repaired systemis returned in the same condition as it was, i.e., the failure rate of the repaired system remains the same as it was just prior tofailure. The minimal repair cost depends on the age and the number of repair, and the cost for corrective replacement is lar-ger than that for preventive replacement. Salvage value for an un-failed system that preventive replaced should be involvedinto the cost model. The cost effectiveness for operating the system in an innite time-span is adopted as a criterion of opti-mality. The problem is to determine the scheduled ordering time so as to maximize the expected cost effectiveness. Undersome reasonable assumptions, the existence and uniqueness of optimum ordering time is derived and discussed. Finally,numerical examples for illustration this problem are demonstrated.

The reason why we considered the system has two types of failures when it fails at age t is that this is a popular approachto model an imperfect repair (maintenance) action in the research area of reliability and maintenance policy. And, such anapproach is much closer to real life situation. As described in Pham and Wang [24] : the repair or maintenance of a deteri-orating system is often imperfect, i.e., the system after repair (maintenance) will not as good as new but younger. In Nak-agawa [25,26] , he treated the imperfect preventive maintenance (PM) in this way: the component is returned to the asgood as new state (perfect PM) with probability p and returned to the as bad as old state (minimal PM) with probabilityq ¼ 1 p after PM. In Brown and Proschan [27] , they consider the following model of the repair process. A unit is repairedeach time it fails. The executed repair is either a perfect repair with probability p or a minimal repair with probability 1 p.Block et al. [28] extended the above imperfect repair model (Brown and Proschan) to the age-dependent case: an item is

repaired at failure. With probability pðt Þ, the repair is a perfect repair; and with probability qðt Þ ¼ 1 pðt Þ, the repair is aminimal repair, where t is the age of the item in use at the failure time. Chien [29] also considered system failure at age t might be one of two failure types. Type I failure occurs with probability qðt Þ and is repaired immediately. Type II failure oc-curs with probability pðt Þ ¼ 1 qðt Þ and requires system replacement.

The reminder of this paper is organized as follows. The ordering policy and model assumptions are described in Section 2.The mathematical formulation is established in Section 3. Based on the model, the optimal ordering policy is derived, and itsstructural properties are presented in Section 4 . Finally, sensitivity analysis is carried out through numerical examples inSection 5 , and some comments are concluded in Section 6 .

2. The ordering policy and assumptions

The spare ordering policy for preventive replacement is according to the following schemes:

1. The original system begins operating at time 0.2. System failure at age t can be one of two failure types. A type I failure occurs with probability qðt Þ and is repaired imme-

diately. A type II failure occurs with probability pðt Þ ¼ 1 qðt Þ and requires system replacement.3. If the type II failure occurs before a pre-specied ordering time T , then the system is shut down and an expedited order is

made immediately at the failure time instant. Otherwise, the regular order is made at time T .4. When the expedited order is made, then the failed system is replaced by the ordered spare as soon as the spare is deliv-

ered. The lead time for delivering that expedited ordered spare is a generally distributed random variable (r.v.) with prob-ability density function (pdf) z ðt Þ, cumulative distribution function (Cdf) Z ðt Þ and nite mean l e .

5. When the regular order is made, then the original system, no matter operating or not, is replaced by the ordered spare assoon as the spare is delivered. The lead time for delivering that regular ordered spare is a generally distributed r.v. withpdf wðt Þ, Cdf W ðt Þ and nite mean l r .

6. After a replacement, the procedure is repeated.

Various costs for operating this ordering-replacement procedure are dened as follows.

Y.-H. Chien, J.-A. Chen / Applied Mathematical Modelling 34 (2010) 716–724 717

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1. The cost of the i-th minimal repair at age t is / ðK ðt Þ; kiðt ÞÞ where K ðt Þ is the age-dependent randompart, kiðt Þ is the deter-ministic part that depends on the age and the number of minimal repair, and / is a positive non-decreasing continuousfunction.

2. The cost for an expedited order is c e , and the cost for a regular order is c r .3. The cost for a corrective replacement is c c , and the cost for a preventive replacement is c p.4. The cost rate resulting from the system shutdown is c d .5. The salvage value per unit time for the residual lifetime of an un-failed system is v s .

Finally, the following hypotheses are required:

1. Replacements are made perfectly and do not affect the system’s characteristics.2. All failures are instantly detected and is repaired instantaneously if it is a type I failure.3. c e P c r > 0, l r P l e > 0 and c c P c p > 0.

3. Mathematical formulation

3.1. Preliminaries

Suppose the non-homogeneous Poisson process f N ðt Þ; t P 0g with intensity r ðt Þ and successive arrival times S 1 ; S 2 ; . . .

describes the failure behavior of a system. The two system failure types are associated at random with the event epochsof the process. At time S n the system has two possible types of failure. The outcome indicator function is:

dn ¼ 0; ðtype I failure Þ with probability qðS nÞ;

1 ; ðtype II failure Þ with probability pðS nÞ:

Let Lðt Þ ¼ P N ðt Þn¼1 dn count the number of type II failures of the system in the interval ½0 ; t . Then M ðt Þ ¼ N ðt Þ Lðt Þ counts thenumber of type I failures, and it can be shown that f Lðt Þ; t P 0g and f M ðt Þ; t P 0g are independent non-homogeneous Pois-son processes with intensities pðt Þr ðt Þ and qðt Þr ðt Þ (see, e.g., Savits [30] ). This is similar to the classical decomposition of aPoisson process for a constant event probability p. Let X 1 denote the waiting time until the rst type II failure, then X 1 ¼ inf f t P 0 : Lðt Þ ¼ 1g. Note that X 1 is independent of fM ðt Þ; t P 0g. Thus, the survival distribution of the time untilthe rst type II failure is given by

F pðt Þ ¼ P ð X 1 > t Þ ¼ P ðLðt Þ ¼ 0Þ ¼ exp

Z t

0

pð xÞr ð xÞdx

: ð1Þ

In order to develop the model, the following lemma from Sheu and Liou [21] is required.

Lemma 1. Let f M ðt Þ; t P 0g be a non-homogeneous Poisson process with intensity q ðt Þr ðt Þ and E½M ðt Þ ¼R t 0 qð xÞr ð xÞdx. Denotethe successive arrival time by S 1 ; S 2 ; . Assume that at time S i (i ¼ 1 ; 2; ) a cost of / ðK ðS iÞ; kiðS iÞÞ is incurred. Suppose that K ðt Þ at age t is a random variable with nite mean E½K ðt Þ. If pðt Þ is the total cost incurred over ½0 ; t Þ, then

E ½pðt Þ ¼ E XM ðt Þ

i¼1

/ ðK ðS iÞ; kiðS iÞÞ" #¼ Z t

0hð xÞqð xÞr ð xÞdx; ð2Þ

where hðt Þ ¼ E M ðt Þ½E K ðt Þ½/ ðK ðt Þ; kM ðt Þþ1 ðt ÞÞ .

3.2. Model development

According to the replacement scheme that described in Section 2, there are three mutually exclusive and exhaustivestates between successive replacements can be dened (see Fig. 1 ):

State 1. A type II failure occurs before the scheduled ordering time T ; i.e., an expedited order is made and the failed systemis corrective replaced by the spare as soon as the spare is delivered.

State 2. A type II failure occurs between T and the arrival of the ordered spare; i.e., a regular order is made and the failedsystem is corrective replaced by the spare as soon as the spare is delivered.

State 3. No type II failure occurs before the arrival of the ordered spare; i.e., a regular order is made and the un-failed sys-tem is preventive replaced by the spare as soon as the spare is delivered.

3.2.1. Expectation of the uptime and downtime in a replacement cycle

According to Fig. 1 , a replacement cycle can be expressed as

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X 1 þ Y e ; if X 1 < T ;T þ Y r ; if T 6 X 1 < T þ Y r ;T þ Y r ; if X 1 P T þ Y r ;

8>:ð3Þ

where Y e and Y r are the random lead time respectively for delivering the expedited and regular ordered spare. Thus, by equa-

tion (3) , the expected cycle length can be obtained as:

Z 1

0 Z T

0ð x þ yÞdF pð xÞdZ ð yÞ þZ

1

0 Z T þ y

T ðT þ yÞdF pð xÞdW ð yÞ þZ

1

0 Z 1

T þ yðT þ yÞdF pð xÞdW ð yÞ

¼ Z 1

0 Z T

0ð x þ yÞdF pð xÞdZ ð yÞ þ Z

1

0 Z 1

T ðT þ yÞdF pð xÞdW ð yÞ

¼ l e F pðT Þ þ l r F pðT Þ þZ T

0F pð xÞdx ¼ l r ð l r l eÞ F pðT Þ þZ

T

0F pð xÞdx: ð4Þ

Since downtime only occurs in the states 1 and 2, thus the expected downtime per cycle is

DðT Þ ¼Z 1

0 Z T

0 ydF pð xÞdZ ð yÞ þ Z

1

0 Z T þ y

T ðT þ y xÞdF pð xÞdW ð yÞ ¼Z

1

0 Z T þ y

T F pð xÞdxdW ð yÞ ð l r l eÞ F pðT Þ: ð5Þ

Furthermore, since the uptime per cycle is the cycle length minus the downtime per cycle, thus the expected uptime percycle can be obtained through Eqs. (4) and (5) , which is given by

U ðT Þ ¼ l r þ Z T

0F pð xÞdx Z

1

0 Z T þ y

T F pð xÞdxdW ð yÞ ¼Z

1

0 Z T þ y

0F pð xÞdxdW ð yÞ: ð6Þ

3.2.2. Expectation of the operational cost in a replacement cycleThe expected cost per cycle is the sum of the cost for ordering, minimal repair, downtime, replacement, and the salvage

value of the system. According to Fig. 1 , it is obviously that the expected cost due to spare ordering per cycle is

c e F pðT Þ þ c r F pðT Þ ¼ c r þ ðc e c r ÞF pðT Þ; ð7Þ

and by (5) the expected cost due to downtime per cycle is

c d DðT Þ ¼ c d Z 1

0 Z T þ y

T F pð xÞdxdW ð yÞ ð l r l eÞF pðT Þ : ð8Þ

For the replacement costs, since the replacement that occurs in the states 1 and 2 is the corrective replacement, and thereplacement that occurs in the state 3 is the preventive replacement, thus the expected cost due to replacement per cycle is

c c F pðT Þ þZ 1

0 Z T þ y

T dF pð xÞdW ð yÞ þ c p Z

1

0 Z 1

T þ ydF pð xÞdW ð yÞ ¼ c p þ ðc c c pÞZ

1

0F pðT þ yÞdW ð yÞ: ð9Þ

For the minimal repair costs, there are three mutually exclusive and exhaustive possibilities for the minimal repair coststhat exist in every cycle, which can be expressed as:

PM ð X 1 Þi¼1 / ðK ðS iÞ; kiðS iÞÞ; if X 1 < T ;

PM ð X 1 Þi¼1 / ðK ðS iÞ; kiðS iÞÞ; if T 6 X 1 < T þ Y r ;

PM ðT þ Y r Þi¼1 / ðK ðS iÞ; kiðS iÞÞ; if X 1 P T þ Y r ;

8>>>:

ð10 Þ

State 1

State 2

State 3

T X


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Thus, by using Lemma 1 and take expectation for the Eq. (10) , the expected cost due to minimal repair per cycle is

Z T

0E X

M ð xÞ

i¼1

/ ðK ðS iÞ; kiðS iÞÞ( )dF pð xÞ þ Z 1

0 Z T þ y

T E X

M ð xÞ

i¼1

/ ðK ðS iÞ; kiðS iÞÞ( )dF pð xÞdW ð yÞþ Z

1

0F pðT þ yÞE X

M ðT þ yÞ

i¼1

/ ðK ðS iÞ; kiðS iÞÞ( )dW ð yÞ ¼Z T

0 Z x

0hðt Þqðt Þr ðt ÞdtdF pð xÞ

þ Z 1

0 Z T þ y

T Z x

0hðt Þqðt Þr ðt ÞdtdF pð xÞdW ð yÞ þ Z

1

0F pðT þ yÞZ

T þ y

0hðt Þqðt Þr ðt ÞdtdW ð yÞ

¼ Z 1

0 Z T þ y

0F pð xÞhð xÞqð xÞr ð xÞdxdW ð yÞ: ð11 Þ

For the salvage value, it seems reasonable that salvage value of a used unit, which is still operable, is proportional to theexpected residual lifetime (Kaio and Osaki [16] ). Because salvage value only occurs in the state 3, thus the expected salvagevalue per cycle is

v s Z 1

0 Z 1

T þ yð x T yÞdF pð xÞdW ð yÞ ¼ v s Z

1

0 Z 1

T þ yF pð xÞdxdW ð yÞ: ð12 Þ

Therefore, by Eqs. (7)–(12) , the expected operational cost per cycle is

C ðT Þ ¼ ðc r þ c pÞ þ ½ðc e c r Þ c dðl r l eÞF pðT Þ þ Z 1

0 Z T þ y

0 F pð xÞhð xÞqð xÞr ð xÞdxdW ð yÞ

þ ðc c c pÞZ 1

0F pðT þ yÞdW ð yÞ þ c d Z

1

0 Z T þ y

T F pð xÞdxdW ð yÞ v s Z

1

0 Z 1

T þ yF pð xÞdxdW ð yÞ: ð13 Þ

Since in our formulation for the spare ordering and replacement process, each replacement is a regeneration point, we canrewrite the cost effectiveness as

s availabilitys expected cost rate ¼

expected uptime per cycleexpected operational cost per cycle

:

Therefore, the cost effectiveness function can be expressed as

CE ðT Þ ¼ U ðT ÞC ðT Þ

; ð14 Þ

where U ðT Þ and C ðT Þ are respectively given by Eqs. (6) and (13) .

4. Optimization analysis

Dene r p; yðt Þ ¼ f pðt Þ=F pðt þ yÞ and F p; yðt Þ ¼ ½F pðt þ yÞ F pðt Þ=F pðt þ yÞ, the following lemma is required and helpful toexamine the existence and uniqueness of the optimum ordering policy.

Lemma 2. Both r p; yðt Þ and F p; yðt Þ are strictly increasing in t if r ðt Þ is strictly increasing in t and p ðt Þ is non-decreasing in t.

Proof. Since r pðt Þ ¼ pðt Þr ðt Þ and r ðt Þ have the same monotone properties, strictly IFR means

r pðt þ yÞ r pðt Þ > 0 ð15 Þ

and

ddt

r pðt Þ ¼ddt f pðt Þ þ r pðt Þ f pðt Þ

F pðt Þ> 0; ð16 Þ

which implies

ddt

f pðt Þ þ r pðt Þ f pðt Þ > 0: ð17 Þ

Thus, by Eqs. (15) and (17) , it yields

ddt

r p; yðt Þ ¼ddt f pðt Þ þ r pðt þ yÞ f pðt Þ

F pðt þ yÞ>

ddt f pðt Þ þ r pðt Þ f pðt Þ

F pðt þ yÞ> 0: ð18 Þ

On the other hand, by Eq. (15) , it yields



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ddt

F p; yðt Þ ¼ F pðt Þ½r pðt þ yÞ r pðt Þ

F pðt þ yÞ> 0: ð19 Þ

Therefore, both r p; yðt Þ and F p; yðt Þ are strictly increasing in t . h

Dene the numerator of the derivative of CE ðT Þ in Eq. (14) divided by F pðT þ yÞ as f ðT Þ; that is

fðT Þ ¼ C ðT Þ U ðT Þ f½ðc e c r Þ c dðl r l eÞr p; yðT Þ þ ½hðT þ yÞqðT þ yÞ þ ðc c c pÞ pðT þ yÞr ðT þ yÞ þ c d F p; yðT Þ þ v sg:

ð20 ÞThen, the main results concerning the optimal ordering time T which maximizes CE ðT Þ is summarized below.

Theorem 1. Suppose that the function ½hðt Þqðt Þ þ ðc c c pÞ pðt Þr ðt Þ is continuous and strictly increasing in t, andðc e c r Þ P c dðl r l eÞ. Then

(i) If f ð0Þ 6 0 , then the optimum ordering time T ¼ 0 , i.e., place a regular order at the same instant when a new system is put inservice and never place an expedited order.(ii) If f ð0Þ > 0 and f ð1Þ < 0 , then there exists a nite and unique optimum ordering time T (0 < T < 1 ) satisfying f ðT Þ ¼ 0 .(iii) If f ð0Þ P 0 , then the optimum ordering time T ¼ 1 , i.e., place an expedited order at the instant of failure and never place aregular order.

Proof. Differentiating CE ðT Þ with respect to T and setting it equal to zero implies f ðT Þ ¼ 0.Further,

ddT

fðT Þ ¼ U ðT Þ ½ðc e c r Þ c dðl r l eÞ ddT

r p; yðT Þ þ

ddT

ð½hðT þ yÞqðT þ yÞ þ ðc c c pÞ pðT þ yÞr ðT þ yÞÞ þ c dd

dT F p; yðT Þ < 0 ð21 Þ

since both r p; yðt Þ and F p; yðt Þ are strictly increasing in t . Thus, the existence of T in the theorem follows trivially. h

Remark 1. A sufcient condition for the optimality in the theorem, ðc e c r Þ P c d ðl r l eÞ, has been widely used in spareordering policies. For example, see Chien [13] , Dohi et al. [14] , Kaio and Osaki [15,16] , Kalpakam and Hameed [17] , Sheuand Liou [21] , and Sheu [20] . However it should be noted that the assumption does not economically justify placing an expe-dited order since the additional cost for the expedition ðc e c r Þ is larger than the savings obtained from the expeditionc dðl r l eÞ. Hence it is meaningful only when there exists such intangibles as loss of goodwill, reputation and credit whichare difcult to be quantied and included in downtime cost.

5. Numerical example and sensitivity analysis

Assume the system’s lifetime is Weibull distribution, which is one of the most common lifetime distributions in reliabilitystudies. The probability density function (pdf) of a Weibull distribution with shape parameter a and scale parameter k is gi-ven by

f ðt Þ ¼ kaðkt Þa 1 exp f ð kt Þa g; t > 0; ð22 Þ

where the parameters are chosen to be a ¼ 2 :0 and k ¼ 2 :0. Assume further that the lead time for delivering an ordered spareis Normal distributed, which is also one of the most common distributions in spare ordering/holding investigation (see Parkand Park [18] and Silver et al. [31] ). In our study, let random variable X N ðl r ; r 2r Þ, the cumulative distribution function(Cdf) of a regular ordered lead time is dened as W ðt Þ ¼ P ð X 6 t j X P 0Þ. Thus, the random lead time will not be less thanzero.

For the purpose of illustration, let us consider the following case. The mean and standard deviation of the lead time fordelivering a regular ordered spare is l r ¼ 0 :5 and r r ¼ 0 :2. The mean expedited ordered lead time is l e ¼ 0:3. Cost param-eters are chosen to be c e ¼ 0 :5, c r ¼ 0 :4, c c ¼ 3 :0, c p ¼ 2 :5, c d ¼ 0:05 and v s ¼ 1 :0. Fig. 2 shows how the expected cost effec-tiveness CE ðT Þ changes with respect to the scheduled ordering time T . The optimal spare ordering time is determined bymaximizing the expected cost effectiveness. Under this case, the optimal policy is placing a regular order at T ¼ 0 :71 andthe corresponding cost effectiveness is CE ðT Þ ¼ 0 :0429.

Furthermore, to investigate the effect of r r on the ordering policy, we have obtained the following by numerical methods.For r r ¼ 0:19 ; 0:20 ; 0:21 ; 0 :22 ; 0 :23, the optimal spare ordering times maximizing the cost effectiveness are 0.88, 0.71, 0.57,0.46, 0.37, respectively; and the corresponding cost effectiveness are 0.0365, 0.0429, 0.0493, 0.0555, 0.0615, respectively. Asmight be expected, as the standard deviation

r r increases, the optimal spare ordering time decreases due to uncertainty; but,

the corresponding optimal cost effectiveness increases, according as r r increases.

Y.-H. Chien, J.-A. Chen / Applied Mathematical Modelling 34 (2010) 716–724 721


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Finally, a more sufcient sensitiveness analysis in terms of cost parameters c e , c r , c c , c p , c d and v s are carried out. Fix someparameters at l e ¼ 0:3, l r ¼ 0 :5 and r r ¼ 0 :2, we vary the remaining parameters to examine their inuence on the optimalsolution, the results are summarized separately in Tables 1–6 . Base on these numerical results, the following observationscan be drawn:

1. From Tables 1 and 2 , the optimal spare ordering time T decreases as the expedited ordering cost c e increase, andincreases as the regular ordering cost c r increase. This is an intuitively results.

2. From Tables 3 and 4 , the optimal spare ordering time T decreases as the corrective replacement cost c c increase, andincreases as the preventive replacement cost c p increase. This is to be expected because when the cost for a failurereplacement is high, one should order a spare more early for preventive replacement to avoid system failures. Conversely,when the cost for a preventive replacement is high, one should wait as long as possible before ordering a spare.

3. From Table 5 , as the downtime cost c d increases, the optimal spare ordering time T decreases. The reason is that when asystemwith higher downtime cost, a spare unit for preventive replacement should be ordered more early to avoid systemshut down.

4. From Table 6 , as the salvage value v s increases, the optimal spare ordering time T decreases. This is also reasonablebecause when the salvage value is large, one should order a spare more early for preventive replacement in order toget the benets from that un-failed unit.

5. From Tables 1–6 , as might be expected, the optimal expected cost effectiveness decreases as the ordering, replacement ordowntime costs increase, and increases as the salvage value increases.

All the numerical results are intuitive and match our expectations.

Scheduled T vs..Cost Effectiveness

0.030

0.035

0.040

0.045

0.050

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

Scheduled T

C E ( T )

Fig. 2. Cost effectiveness as function of scheduled ordering time T .

Table 1

Optimal policies under various c e .

c e T CE ðT Þ

0.5 0.71 0.04291.0 0.61 0.04021.5 0.45 0.03822.0 0.33 0.03682.5 0.27 0.0359

5.0 0.14 0.0338



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6. Conclusion

In this paper we have discussed an optimal spare ordering policy for preventive replacement under the cost effectivenesscriterion. The lifetime of a spare unit is generalized distributed, the lead time for spare delivery is random, and the salvagevalue for the residual lifetime of an un-failed unit is considered. Assuming the failure rate of the system is increasing, wehave obtained some properties regarding the optimal spare ordering time and the corresponding optimal expected costeffectiveness. Theorem 1 presents the conditions for existence and uniqueness of the optimal ordering time that maximizingthe cost effectiveness. A numerical example illustrates the behavior of the optimal solution when selected cost parametersare varied. The results are all intuitive and match our expectations.

It is worth pointing out that, if the unit is inexpensive, quantity purchases might be practical, and it is natural to considera stocking policy to determine how many spares to purchase with each order as Falkner [32] does rather than our simplied

Table 2

Optimal policies under various c r .

c r T CE ðT Þ

0.2 0.62 0.04350.25 0.65 0.04330.3 0.67 0.04320.35 0.69 0.04310.4 0.71 0.0429

Table 3

Optimal policies under various c c .

c c T CE ðT Þ

3 0.71 0.04296 0.70 0.04269 0.68 0.0423

12 0.67 0.042115 0.66 0.0417

Table 4

Optimal policies under various c p.

c p T CE ðT Þ

0.5 0.09 0.08631.0 0.28 0.06511.5 0.44 0.05442.0 0.58 0.04762.5 0.71 0.0429

Table 5

Optimal policies under various c d .

c d T CE ðT Þ

0.01 0.76 0.04340.03 0.73 0.04320.05 0.71 0.04290.07 0.69 0.04270.10 0.66 0.0424

Table 6

Optimal policies under various v s .

v s T CE ðT Þ

0.10 0.84 0.04030.25 0.82 0.04070.50 0.78 0.04140.75 0.75 0.04221.00 0.71 0.0429

Y.-H. Chien, J.-A. Chen/ Applied Mathematical Modelling 34 (2010) 716–724 723


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ordering policy. Thus, the unit in an ordering policy is usually a high-cost complex system. Moreover, it is also need to notethat a spare ordering policy is an important factor in deriving an optimal maintenance scheme. Practitioners should take therandom lead time for a spare delivery into account in making maintenance scheme. Similar analysis to the article presentedhere can be conducted for various combinations of product warranties and maintenance policies to help practitioners makebetter decisions.

Acknowledgements

We are very grateful to the referees for their insightful comments and suggestions which greatly improved the articlesignicantly. All of their suggestions were incorporated directly in the text. This research was supported by the National Sci-ence Council of Taiwan, under Grant No. NSC 97-2221-E-025-004-MY3.

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Optimal Spare Ordering Policy for Preventive Replacement Under Cost