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Available online 5 June 2013

Condition-based spares orderingProportional Hazards ModelLead time

facversy, thts p

measurement of spare performance, based on the stressstrength interference theory; which we have

ucial feness.ction clead t

essential to system performance of these industries.

fcient decisionsntial in the costn efcient sparestrategy can bemes necessary toct the system thering decision-aid

method to secure the spare management performance into an

Contents lists available at SciVerse ScienceDirect

.elsevier.com/locate/ress

Reliability Engineering

Reliability Engineering and System Safety 119 (2013) 199206actions [6]. In addition, maintenance strategy can be treated byrpascual@ing.puc.cl (R. Pascual), p.knights@uq.edu.au (P. Knights).assets corresponds to inventories [3]. Of these assets, critical spare

1.1. Critical spare parts and maintenance strategy

The need for spare parts inventories is dictated by maintenance

0951-8320/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ress.2013.05.026

n Corresponding author. Tel.: +56 982714361.E-mail addresses: drgodoy@puc.cl (D.R. Godoy),Spare parts play a fundamental role in the support of criticalequipment. In a typical company, approximately one third of all

operational environment that needs continuity to compete into ademanding business context.add value and enhance protability of operations is needed byrms in pursuit of performance excellence [1]. An efcientresources ordering is indispensable to achieve signicant avail-ability exigencies of equipment-intensive industries, such asmining, aeronautic, nuclear energy, or defence. This equipment issupported by spare parts inventories, which are particularlyrelevant considering the inuence of stock-outs on downtime[2]. Appropriate spare parts allocation decisions are therefore

ment of costly components, which have extremequipment-intensive industries [5]. Therefore, eabout spare-stocking policies can become essestructure of companies. In order to provide amanagement performance, a suitable orderingrelevant. A spare part classication scheme becoset optimal policies for those spares that may affemost, and at the least effort. We proposed an ordement. This situation affects reliability and, more importantly,system throughput. Continuous improvement of the ability to

tinuity. The improvement of key prots on both logistics andmaintenance performance can be achieved by inventory manage-

ely criticality onCondition-based service levelStressstrength interference modelsOrdering process

1. Introduction

Decision-making processes are cra scenario of intense competitivfrequently required to reduce produutilization, misguided decisions maycalled Condition-Based Service Level (CBSL). We focus on Condition Managed Critical Spares (CMS), namely,spares which are expensive, highly reliable, with higher lead times, and are not available in store. As amitigation measure, CMS are under condition monitoring. The aim of the paper is orienting the decisiontime for CMS ordering or just continuing the operation. The paper presents a graphic technique whichconsiders a rule for decision based on both condition-based reliability function and a stochastic/xedlead time. For the stochastic lead time case, results show that technique is effective to determine the timewhen the system operation is reliable and can withstand the lead time variability, satisfying a desiredservice level. Additionally, for the constant lead time case, the technique helps to dene insurance spares.In conclusion, presented ordering decision rule is useful to asset managers for enhancing the operationalcontinuity affected by spare parts.

& 2013 Elsevier Ltd. All rights reserved.

or organizations withinSince companies areosts and increase asseto over-stress on equip-

parts have special relevance because they are associated with bothsignicant investment and high reliability requirements. As anexample, spares inventories sum up above US$ 50 billion in theairlines business [4]. The mismanagement of spare parts thatsupport critical equipment conduces to considerable impacts onnancial structure and severe consequences on operational con-Keywords:Critical spare partsCritical spare parts ordering decisions uand stochastic lead time

David R. Godoy a,n, Rodrigo Pascual a, Peter Knightsa Physical Asset Management Lab, Department of Mining Engineering, Ponticia Univerb School of Mechanical and Mining Engineering, The University of Queensland, QLD 40

a r t i c l e i n f o

Article history:Received 24 July 2012Received in revised form17 May 2013Accepted 24 May 2013

a b s t r a c t

Asset-intensive companiesscenario often leads to oavailability, reliability, andand operational structuresmanagement of spare par

journal homepage: wwwng conditional reliability

d Catlica de Chile, Av. Vicua Mackenna 4860, Santiago, Chilet. Lucia, Brisbane, Australia

e great pressure to reduce operation costs and increase utilization. This-stress on critical equipment and its spare parts associated, affectingstem performance. As these resources impact considerably on nanciale opportunity is given by demand for decision-making methods for therocesses. We proposed an ordering decision-aid technique which uses aand System Safety

rate is dened as the expected number of units demanded per time

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 199206200Condition-Based Maintenance (CBM). In this case, modelsincorporate information about equipment conditions in order toestimate the conditional reliability. This information comes from,for instance, vibrations measurements, oil analysis, sensors data,operating conditions, among others. These measures are calledcovariates. Covariates may be included on the conditional relia-bility using the Proportional Hazards Model (PHM) [7], whichallows combining age and environmental conditions. In the inter-action between operational environment and equipment, whileage can be relatively easy to notice, deterioration can be measuredby conditions assessment [8]. Therefore, CBM becomes useful toset maintenance policies even with different levels of monitoringrestrictions. Compared to usual time-based maintenance strate-gies, condition monitoring systems offer signicant potential toadd economic value to spares management performance [9].Particularly, this paper uses CBM models to calculate conditionalreliability in order to make ordering or replacement decisions.

Lead time is another important aspect to consider in spareparts ordering. The random time between fault event and theactual component failure may cause system performance dete-riorations [10]. Nonetheless, it also provides a opportunity win-dow to set replacement policies. Logistically, there are also delaysbetween the order of spares and their arrival [11]. This situation iseven more crucial when spare parts are critical, since they are notalways available at the supplier store. Customs delays and theneed of special transport are a source of signicant lead times;moreover, when dealing with complex equipment parts made toorder, lead times may exceed a year [12]. The lack of these itemsbecause of a delay in delivery (and their consequent installation)may have severe consequences in the operational continuity.

The core of this paper is on those critical spare parts that affectproduction, safety, are expensive, highly reliable, and usually areassociated with higher lead times. These items are critical too, whenthey support equipment which is essential in an operational environ-ment [2]. Henceforward, all spare parts that meet these characteristicswill be called, Condition Managed Critical Spares, or just CMS. Fig. 1shows a diagramwith the spare parts that we are focusing on. CMS arerepairable, however their repair times are slower than supplier leadtimes, this particularity turns these CMS into non-repairable spareparts for the purposes of this model. As CMS are not available in store,CMS condition is monitored as a mitigation measure to its criticality inthe operation. Justication for not having them in store lies in theexpectation that the CBM models will predict failures with sufcientlead time to overcome the need for spare-stocking.

Previous works have treated the decision-making process usingCBM, for instance: research deals with a continuously deterioratingsystem which is inspected at random times sequentially chosen withthe help of a maintenance scheduling function [13]. There is alsoresearch obtaining an analytical model of the policy for stochasticallydeteriorating systems [14]. However, spare parts issues are notincluded on those papers. Furthermore, there are several researchesfor CBM policies that consider unlimited spare parts which alwaysare available [15]. Nevertheless, the focus of this paper is on criticalspare parts which are, precisely, not available in store. According to[16], few existing ordering and replacement policies are proposed inthe context of condition-based maintenance. In fact, the workdescribed by [16] aims to optimize CBM and spare order manage-ment jointly. Other works [17,18] consider optimal ordering andreplacement policy of a Markovian degradation system undercomplete and incomplete observation, respectively. However, thedifference between this paper and the works stated above is the needto install a user-friendly technique to decision-making process forasset managers in order to improve the spare parts managementconsidering the unique characteristics of CMS. In accordance withcurrent industrial requirements, a graphical tool of this type could be

easy to implement. Spare parts estimation based on reliability andperiod for an item that can be immediately satised from stock athand. Meanwhile, units in service are the expected number of units inroutine resupply or repair at a random point in time. The work statedby [2] uses the instantaneous reliability of stock term as one of itscriteria for determining an optimal stock level. Instantaneous reliabilityis dened as the probability of a spare being available at any givenmoment in time. This measurement can be equivalent to ll rate. Inspite of these valuable denitions, the spare part reliability conceptused in this paper is signicantly different. The source of thisdistinction is given by the critical nature of spare parts which areconsidered in this paper, specially its uniqueness characteristic.Usually, these kinds of critical spare parts are not available in store,thus a common concept such as ll rate is not completely applicable.

For the latter reason, it seems appropriate to introduce a newconcept which we have called as Condition-Based Service Level(CBSL). CBSL is based on the stressstrength interference theory [21].This theory considers two main variables: a stress which is any loadapplied on a system and that may produce a failure (in this case,depletion on service level), and a strength which is the maximumvalue that system can withstand without failing. Therefore, CBSL isdened as the probability that the stress does not overcome thestrength. Stressstrength interference models are widely applied incomponent reliability analysis [22]. Due to the model ability to beused when probability distributions are known and, also, both stressand strength could be general in meaning [22], it is possible to adapta version. For purposes of this paper, stress can be represented bylead time and strength by conditional reliability.

The paper presents a graphic method which uses CBSL as keyindicator, to achieve an effective policy to dene the suitable timefor CMS ordering, through a rule decision based on bothcondition-based reliability function and a stochastic/xed leadtime. The aim of the paper is orienting the decision about CMSordering or to continue the operation process without ordering.The reliability threshold can be chosen by each company accordingto their own needs of service level.

Having introduced the importance of CMS ordering process inthe context of operational continuity, the rest of the paper isorganized as follows: Section 2 indicates the model formulation.Section 3 shows the associated case study. Finally, Section 4reveals the conclusion of the work.

2. Model formulation

In order to precise the CMS ordering decision-making, it isnecessary to estimate the conditional reliability and add theinuence of lead times. The following items dene the calculationmethodology of these aspects.

2.1. Conditional reliability model

For the sake of self-containment, we describe in detail relevantelements which are developed in [23,24]. Reliability function is basedenvironment-operational conditions is a method to improvesupportability. This method can guarantee non-delay in spare partslogistics and to improve production output [19].

1.2. Spare management performance: condition-based service level

There are several denitions to measure spare managementperformance. According to [20] three obvious indicators are readyrate, ll rate, and units in service. Ready rate is the probability that anitem observed at a random point in time has no back orders (backorder is considered as any demand that cannot be met from stock). Fillprimarily on the Markov Failure Time Process model. The reliability

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 199206 201function of an item can be dened as the probability of survival aftera certain interval of time t. For the conditional case (namely,assuming that item has been operated by a time x), the probabilityof interest is PT4tjT4x, where T is the equipment lifetime. Thisreliability is interesting in CBM, given that it is assumed that the itemhas been operated until the inspection moment [2]. It is assumedthat hazard rate can be incorporated into the model using PHM, e.g.[7,25,26]. This method is widely accepted in order to incorporatecondition data of equipment [2] (as used in CBM). Hence

t t; Zt 0teiiZit 1

where 0t is a baseline hazard rate, while i is the weight of eachtime-dependent covariate Zi(t). For the present paper, a Weibull-PHMwas used, thus the hazard rate is

t t; Zt

t

1eiiZit; t0 2

The use of Non-Homogeneous Markov Process (NHMP) is ofparticular interest in applications of CBM [2,24]. Transition prob-abilities (from a state i to a state j, of covariates under study) canbe dened as

Lijx; t PT4t; Zt jjT4x; Zx i; xt 3

where T is a random variable representing the failure time of theitem. Note that covariates Z(x) can be discretized within intervalsusing values ranges for a nite number of states: 0;1;2;; s. Forexample, if the condition under study is the oil level of a motor,the states could be described through intervals with levels limitsas: low, normal, and dangerous.

Fig. 1. Condition manatraobt

tim

Rt

Lxbe

the

tL

tw

(i)

gedTnaT

j

I;

uT

L

rDtp

Io

chus, it is possible to nd a relationship between hazard rate andsition probabilities Lijx; t [24]. The reliability of interest can beined combining both concepts (conditional probability and PHM).he reliability at time t, given that the spare has survived until ae x, and at that time x the condition is Zx i, is given byx; i PT4tjT4x; Zx i

jLijx; t; xt 4

f the matrix Lx; t Lijx; t is dened and it is assumed thatx I (where I is the identity matrix), the Markov property cansed [24].hen, it is demonstrable that all functions Lijx; t; xt satisfysystem of equations [23]:

x; t Lx; tLt Lx; ttDt 5

et

t ijx is the matrix of transition rates. The transitionates ijx can be estimated using the approach of [23].t t; iij is a diagonal matrix, with ij pijx; x (i.e., itakes the value 1 when i j, and the value 0 when ij). For thisarticular case, Dt =t=1eiiZitij.

n order to solve the system of equations described in Eq. (5),cases are identied [24]:

Case I: If the failure rate is only a function of the conditionprocess, namely t gZt. Then, the solution is given by

L0; t eDt 6

ritical spares.

timtim(ii)(str

2.2.1I

timequlead

CBS

3. Case study

3.1. Condition-based reliability

The following case is an adapted version of a case studydescribed by [25]. The spare of interest is an electric motor of amining haul truck and, based on expert judgement, oil is the keyfactor to model the condition process. Table 1 describes the modelparameters. Covariates were discretized in three bands, as shownin Table 2. In addition to, Table 3 indicates the estimated matrix oftransition probabilities.

The conditional reliability function can be estimated for the 3different initial states of oil by using the methodology indicated inSection 2.1. As an example, Fig. 4 shows the conditional reliability

Fig. 2. CBSL as overlap of conditional reliability and lead time.

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 199206202Ftail

anae considering two cases described by [21]: (i) stochastic leade (stress) and stochastic conditional reliability (strength), andconstant lead time (stress) and stochastic conditional reliabilityength).

. Stochastic lead time and stochastic conditional reliabilityf both variables are stochastic, CBSL is the probability that leade (stress) is less than conditional reliability (strength). Orivalently, the probability that conditional reliability exceedstime. As a result, CBSL is given by

L Pxy Z 0

Z y0

f xx dx

f yy dyZ 0

Fxyf yy dy

11ig. 2 exhibits the CBSL which is dened by the area where bothcurves overlap or interfere with each other. This interference(ii) Case II: If the failure rate is a function of age and currentcondition state, namely: t gt; Zt. Then, the solution canbe approximated for the following method called product-property.

Lk;m m1

i k~Li 7

where

~L k eR k1k

Dt dte 8

denes the approximation interval length, such thatkxk 1 and m1t m, kom [2]. In general, whilethe value of is smaller, the precision of the reliabilityestimation is better [24] (but also the amount of iterationswill be larger).

The stated Case II corresponds to Weibull-PHM model used inthis paper, because of it depends on age and condition. Then, theproduct-property method was used in order to estimate thematrix of transition probabilities, and consequently, the reliabilityfunction. In [24] also is dened other method to estimate thesolution of the system of Eq. 5 called product-integral method.However, the product-property method is more accurate thanthe product-integral method when larger values of are used.Besides, the product-property method is convenient because itrequires the estimation of only one transition matrix.

2.2. Condition-based service level (CBSL)

CBSL could be estimated adapting the structure given by stressstrength interference theory [21]. Let x be the stress randomvariable and f(x) be its probability density function. Likewise, lety be the strength random variable and f(y) be its probabilitydensity function. Therefore, the probability that stress does notexceed an x0 value is

Pxx0 Fxx0 Z x00

f xx dx 9

Also, the probability that strength does not exceed an y0 valueis

Pyy0 Fyy0 Z y00

f yy dy 10

In order to calculate CBSL, this paper recognizes lead time asequivalent to stress and conditional reliability as equivalent tostrength. While conditional reliability is estimated using themodel described in Section 2.1, the work adds the effect of leadlysis between stress and strength is the reason of the theory name.2.2.2. Constant lead time and stochastic conditional reliabilityIf the lead time is a known constant value xs and conditional

reliability is a random variable, then CBSL is the probability thatconditional reliability exceeds the constant lead time. Hence

CBSL Pyxs Z xs

f yy dy 12

Consequently, this instance could be considered as a specialcase of when both stress and strength are stochastic.

2.3. Spare part ordering decision rule

Considering a deterministic lead time L, spare part orderingtime To is dened by time Tth at which desired reliability thresholdRth is reached. Note that often a constant lead time is not the caseand variations on the delivery time exist [25]. Companies canchoose several scenarios of reliability threshold in order to obtaina given service level. Therefore, the decision rule can be describedby

To infft0 : LTthg 13Fig. 3 illustrates this rule. It uses data from a numerical example

given by [24]. If lead time L is less than Tth at a given inspectiontime t (case shown by L1) then equipment can continue operatingwith the same spare part, because there is enough time until thearrival of a new spare faced with a potential need (because ofthe policy to keep reliability Rth). If lead time L is greater than Tth(case shown by L2) then an ordering decision is required, other-wise spare part will not be able to ensure the operationalcontinuity of the equipment supported by the spare-stocking. Iflead time L and Tth are equal then an ordering decision must bealso made, because this situation is likely to require a setup timefor the new spare part.x , y

Pro

babi

lity

Den

sity

Fun

ctio

n

f(x) Lead Time (Stress)f(y) Conditional Reliability (Strength)for State 0. Working ages have been set in operational hours (h).

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 199206 2030 5000 10000 150000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Working Age (h)

Rel

iabi

lity

Tth

Rth

CBSLReliability Threshold

L2

L1Conditional reliability is tted with a Weibull distribution fordifferent initial survival times (t0 0 (h) or 0 (months),t0 10;920 (h) or 12 (months), and t0 21840 (h) or 24 (months).Fig. 5 displays the model t. A KolgomorovSmirnov test wasapplied to prove the model and the results were satisfactory.

Having set the Weibull distribution for conditional reliabilityand considering the CBSL denition, the two cases mentioned inSection 2.2 are tested. Firstly, the case where both lead time(stress) and conditional reliability (strength) are stochastic. Sec-ondly, the case where conditional reliability is stochastic, but leadtime is constant.

3.2. Condition-based service level considering stochastic lead time

We test using different distributions for lead time (includingconstant lead time in next section). In this sense, the aim isdetermining the capability to withstand lead time variability. The

Fig. 3. Spare part ordering decision rule.

Table 1Baseline hazard rate parameters.

Parameter Value Units

3.5 19,003 (h) 0.0001742 (ml/particles)

Table 2Oil initial system states and covariate bands.

Initial state Covariate band (ml/particles) State value (particles/ml)

State 0 (053.73) 7State 1 (53.7387.91) 76.5State 2 87:91 11,586

Table 3Transition probabilities for motor condition.

j 1 2 3

p1-j 0.99797 0.00202 0.00001p2-j 0.00159 0.99832 0.00009p3-j 0.00317 0.00181 0.995050 1 2 3x 104

0

0.2

0.4

0.6

0.8

1

Working Age (h)

Con

ditio

nal R

elia

bilit

y

State 0

Fig. 4. Conditional reliability function at oil initial state.

0.6

0.8

1

elia

bilit

ychoice of any lead time distribution is dened by deliveryconstraints of spare part supplier. Fig. 6 exhibits four distributionswhich are considered to t lead time, namely: exponential,truncated normal, Weibull with 2 parameters (Weibull (2p)), andWeibull with 3 parameters (Weibull (3p)). The same mean is setfor all distributions. With an estimated operational utilization of80%, 1 operational week is equivalent to 210 operational hours (h).In this case, mean lead time is set at 2730 (h) (equivalent to 13weeks or 3 months).

Fig. 7 exhibits a performance realization as a result of evolutionover time (t) because of interaction between conditional reliabilityand lead time. Conditional reliability has been tted as a Weibull(2p) distribution. Fig. 8 is a top view of the same realization whichillustrates that CBSL (probability that strength is greater thanstress) is declining as the conditional reliability decreases. In otherwords, component is becoming older over time because theevolution of condition, thus service level is also declining.Figs. 912 show the values of CBSL for different scenarios of meanlead time and for different initial survival times. Standard devia-tion depends on each distribution.

If CBSL is greater than a given reliability threshold Rth, then thesystem is able to resist stress satisfying the desired service level.Thus, equipment can continue operating and a spare part order isnot necessary. On the other hand, if CBSL is less than Rth, then aspare part order is mandatory because of spare part will not beable to withstand the lead time variability, and the desiredreliability would not be accomplished.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

0.2

0.4

Working Age (h)

Con

ditio

nal R

Weibull (=3.5,=18649)t0=0 (h)

Weibull (=1.7,=8324)t0=10920 (h)

Weibull (=1.2,=3025)t0=21840 (h)

Fig. 5. Weibull model t for conditional reliability at different initial survivaltimes.

0.6

sed

Ser

v

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 1992062044

5

6

7

8 x 104

Mean Lead Time=2730 (h)

ensi

ty F

unct

ion

Exponential(=1/2730)Truncated Normal(=2730, =546)Weibull (2p)(=1.5, =3024)Weibull (3p)(=1.5,=2016, t0=910)3.3. Condition-based service level considering constant lead time(special case)

Table 4 evidences the decision-making for different reliabilitythresholds and their respective working ages, where spare partordering depends on the decision rule (Section 2.3). In the currentcase, three threshold values are considered, namely: 99%, 95%, and90%. Three lead time scenarios in order to realize the effect ofthem on service level are considered. Markov reliability model

0 1000 2000 3000 4000 50000

1

2

3

Lead Time (h)

Pro

babi

lity

D

Fig. 6. Probability density functions versus lead time.

0 0.5 1 1.5 2 2.5 3x 104

0

20

40

0

0.5

1

1.5

2

2.5

x 104

t

x , y

Pro

babi

lity

Den

sity

Fun

ctio

n

PDF Lead TimePDF Conditional Reliability

Fig. 7. Performance realization of stress versus strength.

Fig. 8. Top view of CBSL for a given realization.0.8

1

ice

Leve

lallows setting any initial condition for spares. In this regard, acomplete range of practical operational environments can berepresented. For example, if it is assumed that the motor is new,then the initial condition is as-good-as new (State 0).

As expected, when lead time is increasing, reliability con-straints are more demanding and ordering decision time turns

0 0.5 1 1.5 2 2.5x 104

0

0.2

0.4

Working Age (h)

Con

ditio

n B

a

ExponentialTruncated NormalWeibull (2p)Weibull (3p)Constant

Fig. 9. CBSL for initial survival time of 0 months.

0.5 1 1.5 2 2.5x 104

0

0.2

0.4

0.6

0.8

1

Working Age (h)

Con

ditio

n B

ased

Ser

vice

Lev

el

ExponentialTruncated NormalWeibull (2p)Weibull (3p)Constant

Fig. 10. CBSL for initial survival time of 3 months.

1 1.5 2 2.5x 104

0

0.2

0.4

0.6

0.8

1

Working Age (h)

Con

ditio

n B

ased

Ser

vice

Lev

el

ExponentialTruncated NormalWeibull (2p)Weibull (3p)Constant

Fig. 11. CBSL for initial survival time of 6 months.

sooner. As shown in Table 4, for a reliability threshold of 95%, thedecision changes from continue without ordering to order thespare, when the lead time increases from 847 (h) to 3388 (h). Onthe other hand, initial condition states also play a role. For a leadtime of 1694 (h) and at the same reliability threshold of 95%, agreater deterioration level makes to change the decision fromcontinue to order.

Working age of this decision map makes practical sense untilapproximately 6 months of expected lead time. If lead time isgreater than that period, then it could be better to stock spareparts. However, CMS are unique and are not backed-up. Therefore,another important factor is when the component becomes older.Fig. 13 demonstrates this situation, considering different scenariosof to a situation of increasing aging. Table 5 displays the samedecision map but considering 2=3 from the original value.

Using new , the situation becomes critical at a lower time.With the original model, the scenario of always ordering hap-pened just at 6 months. With new , the decision of alwaysordering should be already made with any initial state at3 months. This is relevant because if lead time is 6 months, spareparts should be purchased as soon as the component starts tooperate. Then, with this map is also possible to visualize thoseparts that can be classied as insurance spares.

4. Conclusions

This work provides a technique to enhance spare parts orderingdecision-making when companies need to ensure a reliabilitythreshold restricted by a lead time. Case study showed thatcondition data could be an accurate indicator of component stateaffecting the shape of reliability function. The ordering process canbe affected by different initial survival times and initial conditionstates; they can change the decision for same reliability thresholdeven. On the other hand, lead time is a relevant factor in orderingdecision. The ordering policy is sensitive to different scenarios oflead times; they can also modify the spare part ordering decision ifthe aim is ensuring the operational continuity of the equipmentsupported by the spare part stock. It was concluded that, in orderto fulll with operational continuity, condition data can be apowerful tool for including in spare management. The need offocus on critical spares and severe consequences on equipmentperformance, demands a friendly technique which can be used inan environment where decisions are needed quickly, in thisregard, the presented decision rule is easy to implement graphi-cally and it can be used by asset managers to enhance operationalcontinuity.

Future works may include a combination of CBSL with asso-ciated maintenance and logistics costs in order to obtain a globalperspective of asset management.

1.5 2 2.50

0.2

0.4

0.6

0.8

1

Con

ditio

n B

ased

Ser

vice

Lev

el

ExponentialTruncated NormalWeibull (2p)Weibull (3p)Constant

seve

0.1

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1C

ondi

tiona

l Rel

iabi

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1(2/3)(1/2)(1/3)

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 199206 205x 104Working Age (h)

Fig. 12. CBSL for initial survival time of 12 months.

Table 4Motor ordering decision for different reliability thresholds, considering

Reliability threshold (%) Expected time (h)

State 0 State 1

Expected lead time910 (h) (1 month).99 494 46795 1753 172390 3100 3074

Expected time to order (h)Expected lead time2730 (h) (3 months)99 494 46795 1753 172390 3100 3074

Expected time to order (h)Expected lead time5460 (h) (6 months)99 494 46795 1753 172390 3100 3074ral lead time scenarios.

Ordering decision

State 2 State 0 State 1 State 2

122 Order Order Order1067 Continue Continue Continue2518 Continue Continue Continue

122 Order Order Order1067 Continue Continue Order2518 Continue Continue Order

122 Order Order Order1067 Order Order Order2518 Order Order Order

0 1 2 3 4x 104

0

Working Age (h)

Fig. 13. Conditional reliability considering aging by depletion of .

Table 5Motor ordering decision for different reliability thresholds, considering several lead time scenarios.

Reliability threshold (%) Expected time (h) Ordering decision

State 0 State 1 State 2 State 0 State 1 State 2

Expected lead time910 (h) (1 month).99 284 283 227 Order Order Order

917 Continue Continue Continue1691 Continue Continue Continue

D.R. Godoy et al. / Reliability Engineering and System Safety 119 (2013) 199206206Acknowledgements

The authors wish to acknowledge the partial nancial supportof this study by the FOndo Nacional de DEsarrollo Cientco YTecnolgico (FONDECYT) of the Chilean government (project1130234).

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227 Order Order Order917 Order Order Order

1691 Order Order Order

227 Order Order Order917 Order Order Order

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Critical spare parts ordering decisions using conditional reliability and stochastic lead timeIntroductionCritical spare parts and maintenance strategySpare management performance: condition-based service level

Model formulationConditional reliability modelCondition-based service level (CBSL)Stochastic lead time and stochastic conditional reliabilityConstant lead time and stochastic conditional reliability

Spare part ordering decision rule

Case studyCondition-based reliabilityCondition-based service level considering stochastic lead timeCondition-based service level considering constant lead time (special case)

ConclusionsAcknowledgementsReferences