spare pats demand 2011

Transportation Research Part E 47 (2011) 1194–1209

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Spare parts demand: Linking forecasting to equipment maintenance

Wenbin Wang a,b, Aris A. Syntetos a,⇑a Centre for Operational Research and Applied Statistics, University of Salford, UKb School of Economics and Management, University of Science and Technology of Beijing, China

a r t i c l e i n f o

Article history:Received 19 September 2010Received in revised form 30 December 2010Accepted 31 March 2011

Keywords:Spare partsMaintenanceForecastingInventory managementDelay time

1366-5545/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.tre.2011.04.008

⇑ Corresponding author. Address: Centre for OR anUK. Tel.: +44 161 295 5804.

E-mail address: [email protected] (A.A. Sy

a b s t r a c t

Demand for spare parts is typically intermittent and forecasting the relevant requirementsconstitutes a very challenging exercise. Why is the demand for spare parts intermittent andhow can we use models developed in maintenance research to forecast such demand? Weattempt to answer these questions; we present a novel idea to forecast demand that reliesupon the very sources of the demand generation process and we compare it with a well-known time-series method. We conclude that maintenance driven models are associatedwith a better performance under certain conditions. We also outline an inter-disciplinaryagenda for further research in this area.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Service spare parts are ubiquitous in modern societies. Their need arises whenever a component fails or requires replace-ment. In some sectors, such as the aerospace and automotive industries, a wide range of service parts are held in stock, withsignificant implications for equipment performance and inventory holding. Their management is therefore an important task(Boylan and Syntetos, 2008).

A distinction should be drawn between preventive maintenance and corrective maintenance. Demand arising from pre-ventive maintenance is scheduled and is stochastic with regards to the demand size but deterministic as far as the demandarrival is concerned. Demand arising from corrective maintenance, after a failure has occurred, is stochastic with regards tothe time arrival but deterministic in quantity (being one in most cases). Both require forecasting on the part of the stockistwho is holding the relevant part. Such demand structures are typically intermittent in nature, meaning that demand arrivesinfrequently with (many) time periods showing no demand at all. In addition, demand when it occurs is not necessarily for asingle unit, a very low demand size (slow demand) or a ‘constant’ requirement (clumped demand). That is to say, demandsizes may be highly variable leading to what is termed as ‘lumpy’ demand. In Fig. 1, the demand for two spare parts from theRoyal Air Force (RAF, UK) is graphically depicted. In both cases demand is intermittent but is associated with differing pro-files of the demand size distribution.

Intermittent demand patterns are very difficult to deal with from a forecasting (and stock control) perspective because ofthe associated dual source of variation (demand arrivals, or correspondingly inter-demand intervals, and demand sizes).There have been a number of considerable advancements in this area in the recent years, all of which though have beenmainly focusing on coping, reactively, with the compound nature of the demand patterns under concern. However, no at-tempts have been made to characterise the very sources of such demand patterns for the purpose of developing more effec-

. All rights reserved.

d Applied Statistics, Salford Business School, University of Salford, Maxwell Building, Salford M5 4WT,

ntetos).

http://dx.doi.org/10.1016/j.tre.2011.04.008

mailto:[email protected]

http://dx.doi.org/10.1016/j.tre.2011.04.008

http://www.sciencedirect.com/science/journal/13665545

http://www.elsevier.com/locate/tre

0

10

20

30

40

50

60

70

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecTime period

Dem

and

(Uni

ts)

Slow demand Lumpy demand

Fig. 1. Intermittent demand.

W. Wang, A.A. Syntetos / Transportation Research Part E 47 (2011) 1194–1209 1195

tive, pro-active, mitigation mechanisms. Such an approach would require taking a step back and looking at the industrialmaintenance processes that generate the relevant demand patterns. This is precisely what we do in this paper. The followingquestions are addressed. Why is demand for spare parts intermittent? How can we use models developed in maintenanceresearch to forecast the demand for spare parts based on the requirements for both corrective and preventive maintenance?We attempt to answer these questions by comparing demand forecasts obtained from a statistical time-series forecastingmethod and maintenance-based methods, using simulated data from a well established maintenance model.

The remainder of our paper is structured as follows. In Section 2 we review the relevant literature and elaborate on theresearch questions of our work. Sections 3–5 describe the simulation experiment developed for the purposes of our research.The results obtained are analysed in Section 6. We conclude, in Section 7, with the implications of our work and some naturalextensions for further advancing knowledge in this area.

2. Research background

There are two fundamental types of maintenance: scheduled or preventive maintenance and unplanned repair based onfailures (corrective maintenance). For preventive maintenance, the timing is usually known in advance, but the demand forspare parts is stochastic. For such operations, part of the demand for spare parts may be known if previous condition mon-itoring or inspection revealed evidence of potentially defective items, or if an age or block based replacement policy is ap-plied. However, in the majority of cases, inspections that are carried out may lead to further replacements. As such, demanddue to preventive maintenance is still stochastic and therefore it needs to be forecasted. An illustrative example is that ofDomestic cars’ service. Before a car arrives, the servicemen do not know which parts should be replaced. Unless a forecastis available and a safety stock decision has been made, new parts may need to be ordered. It may obviously take several daysfor those parts to arrive and this would extend the downtime and, consequently, the cost due to preventive maintenance. Forunplanned repairs/replacements, the consequences of stockouts may include further production downtime with significantcosts. Therefore, for both planned and unplanned maintenance, some kind of safety stock policy is required. Such a policywould also typically consider, implicitly (as part of the inventory holding charges) or explicitly the cost of obsolescence.Replenishment decisions are then calculated according to a probability distribution of the demand, the parameters of which(typically mean and variance) are estimated through a forecasting procedure. In this paper we focus on the issue of forecast-ing mean demand and not on the replenishment related aspects of the problem. A combined spare part inventory and main-tenance model is to be addressed in the next steps of our research. Further consideration of the transportation relatedaspects of the problem is also an area that has not received sufficient attention in the literature. The integration of inventorymanagement and transportation mode selection for spare parts logistics systems (e.g. Kutanoglu and Lohiya, 2008) consti-tutes an interesting avenue for further research.

2.1. Spare parts: forecasting for inventory management

Unless historical data on explanatory variables is available, time-series methods are used to forecast spare parts’ require-ments. Most time-series applications to forecasting intermittent demand rely upon some sort of Croston type methodology.Croston (1972) proposed a method that captures the compound nature of the relevant demand patterns (i.e. demand arrivalsand demand sizes). In particular he suggested using Single Exponential Smoothing (SES) for separately forecasting the inter-val between demand incidences and the demand sizes, when demand occurs. The ratio of the latter over the former may thenbe used in order to estimate the mean demand per time period.

1196 W. Wang, A.A. Syntetos / Transportation Research Part E 47 (2011) 1194–1209

Syntetos and Boylan (2001) showed that Croston’s estimator is biased and in a follow-up paper proposed a bias-adjustedmethod (Syntetos-Boylan Approximation, SBA, Syntetos and Boylan, 2005), in which Croston’s estimates are deflated by afactor of 1 � a/2, where a is the smoothing constant used to update the SES estimates of the mean inter-arrival time for de-mands. The forecasts of mean demand are updated only if demand occurs; else they remain the same. Other adjustment fac-tors that overcome the bias of Croston’s approach have also been discussed in the literature by Boylan and Syntetos (2003),Shale et al. (2006), Teunter and Sani (2009). In addition, it is also true that methods such as Simple Moving Average (SMA)and SES (that do not build estimates from constituent elements) may also be associated with a robust performance. SBA hasbeen shown in a number of independent studies (e.g. Eaves and Kingsman, 2004; Gutierrez et al., 2008; Syntetos and Boylan,2005) to outperform, overall, Croston’s estimator but also other methods that are used in an intermittent demand context. Assuch, it constitutes the benchmark time-series approach (see Syntetos et al., 2009) to be used for forecasting demand in ourstudy. The method is further discussed in more detail in Section 5.

Willemain et al. (2004) presented a non-parametric alternative for forecasting intermittent demands. This approach reliesupon the reconstruction of the empirical distribution through a bootstrapping procedure. Such a procedure renders the esti-mation of the parameters of a hypothesised distribution redundant. The researchers claimed significant improvements inforecasting accuracy achieved by using their approach over Single Exponential Smoothing and Croston’s method. Neverthe-less, further empirical evidence is required in order to develop our understanding of the benefits offered by such a non-parametric approach. Other bootstrapping methods for forecasting intermittent demands have also been discussed by Porrasand Dekker (2008) and Teunter and Duncan (2009).

Overall, research in the area of forecasting for intermittent demand items has developed rapidly in recent years with newresults implemented into software products because of their practical importance (Fildes et al., 2008). Similar developmentshave been contributed to the stock control literature. Many new stock control policies (or modifications to existing ones,such as the order-up-to level) have been formulated that aim at capturing the compound nature of intermittent demand pat-terns. While practical implementations lag considerably behind theoretical propositions in this area, the relevant studieshave certainly advanced knowledge and enabled insight to be gained into operational issues.

Nevertheless, all such studies share a common characteristic: they attempt to provide the best possible modelling of theunderlying demand characteristics without questioning the demand generation process itself. Studying the demand gener-ating process itself could help moving away from the re-active nature of current inventory management procedures for spareparts to more pro-active (in nature) mitigation mechanisms. Given the intuitive appeal of such an approach it is surprisingthat it has not been advanced in the academic literature. As discussed in the previous section the main research questions ofthis work are the following: ‘Why demand for spare parts is intermittent?’ and ‘How we may forecast the demand for spareparts based on a model developed in a maintenance context rather using time series approaches only?’ Before we attempt toanswer these questions, the problem area is also approached from the maintenance research literature.

2.2. Spare parts: maintenance research

Spare parts inventories exist for serving the needs of maintenance and those related to the replacement of plant items.The demand for spare parts depends on the type of maintenance interventions and the failure characteristics of the plantitem concerned. In a typical maintenance setting two types of costs and downtimes are considered, namely, the downtimeand cost due to corrective maintenance and the same due to preventive maintenance. Extensive efforts have been put intothe optimisation of preventive maintenance intervals, replacement schedules and reliability improvements. See Cho andParlar (1991), Dekker (1996), Nicolai and Dekker (2008), Thomas (1986), Wang (2002) for a series of review papers on thissubject. It is noted however that most researches have treated maintenance as an area of research on its own, and did notconsider the impact of the availability of spare parts on the plant downtime and cost due to maintenance. There are someexceptions; in particular for age based and block based maintenance policies (Scarf and Deara, 2003), inventory policies havebeen jointly considered with maintenance-related issues (Armstrong and Atkins, 1996; Brezavšcek and Hudoklin, 2003; deSmidt-Destombes et al., 2007, 2009; Kabir and Al-Olayan, 1994, Kabir and Al-Olayan, 1996). Nevertheless, the replacementsresulting from both those policies compare unfavourably to condition or inspection based replacements in practice (Wang,2008b).

There have been several papers addressing the problem of a failure based repair policy and its connection with spare partsprovision (Albright and Gupta, 1993; Dhakar et al., 1994; Kim et al., 1996; Simpson, 1978; Yeralan et al., 1986). Attention hasbeen paid to how equipment failures impact on the spare parts inventory policy. There are also a few review papers con-cerned with both spare parts inventory and maintenance (see e.g. Kennedy et al., 2002; Nahmias, 1981; Rustenburg et al.,2001). However, no research has been conducted on forecasting spare parts demand from a maintenance based modeland this is part of the focus of this paper. In particular, we seek a forecasting method that is based on regularly planned pre-ventive maintenance activities as well as corrective maintenance activities using a modelling concept called delay time. Theplanned maintenance is not scheduled by an age or block based replacement policy; rather, we look at a general case wherethe need for spare parts is driven by the result of inspections. This is perhaps the most common scenario that one mayencounter in practice (see, for example, Christer, 1999).

To clarify the objective of the type of inspection modelling we are concerned with here, consider a plant item with anperiodic inspection practice every t periods (weeks, months, . . .) with repair of failures undertaken as they arise. Theinspection consists of a check list of activities to be undertaken and a general inspection of the operational state of the plant.

0 u s time

h

Fig. 2. Illustration of the delay time concept, (0, u) refers to the first stage and s � u = h is the delay time. s defect arrives, d failure.

t 2t time3t 4t

Fig. 3. Defective items’ arrival and failure process subject to inspections and replacements at times t, 2t,. . . .

Demand

Time t 2t 3t 4t

2

1

Fig. 4. Demand pattern generated from the process described in Fig. 3.


Any defective items identified leads to immediate repair or replacement and the objective of the inspection is to minimiseoperational downtime. Other objectives could be considered, for example cost, availability or output. For now we focus onthe inspection practice outlined above using the delay time inspection modelling technique.

The delay time modelling of plant inspection and replacements has been discussed in the literature by many authors(Christer and Waller, 1984; Christer, 1999; Jones et al., 2009; Wang and Christer, 2003; Wang, 2008a) with many case basedstudies reported as well by Akbarov et al. (2008), Baker and Wang (1991), Christer et al. (1995, 1997) and Pillay et al. (2001).The delay time concept defines a two stage-failure process: in the first stage the defect arises; the second stage covers theperiod from the moment that the defect has aroused to failure (see Fig. 2).

If we have a number of identical components/items installed and inspected at a regular interval t, then we could have asituation like the one depicted in Fig. 3. These items fail independently since they may be installed in different machines orvehicles.

Further, if we assume that all defective items are maintained by replacement when they are identified to be defective,then we end up with a demand pattern that resembles that depicted in Fig. 4.

Fig. 4 clearly shows that the demand is intermittent (and lumpy) at the time of inspection and replacement. Three factorsinfluence the demand pattern: (i) the initial time distribution of u, (ii) the delay time of h, and (iii) the inspection interval of t.In particular if we have more inspections there will be more preventive replacements and if the delay time is long then wemay have more lumpy demand generated at the time of preventive maintenance. In the next section, we will generate a setof data using the above delay time modelling concept and then seek to forecast the demand by using the SBA statistical timeseries estimation procedure developed by Syntetos and Boylan (2005). The forecast results are to be compared to thosedeveloped using the delay time model discussed by Wang (2008a).

3. Simulation study

3.1. Block based inspection

We start with a simple case where a block based inspection scheme is implemented, i.e. all items concerned are inspectedat a fixed interval regardless of their age. This is typically the case when the items to be inspected constitute part of a largersystem (Wang et al., 2010). Practical examples include production lines, gas turbines, offshore platforms and commercialvehicles. In our simulation study we assume that there are a number of identical components installed in the system. Thefailure process is simulated using the delay time concept discussed earlier. Both the time-series forecasting method andthe delay time maintenance – based model to be shown later are used in order to forecast the demand. Forecasts are then


to be compared to the actual demand data generated by the simulation. This will shed light on the extent to (circumstancesunder) which spare parts’ demand can be forecasted by appropriate maintenance models based on reliability characteristicsand the comparative (dis)advantages of such an approach to time series based forecasting methods.

The assumptions used in our block based inspection scheme are as follows:

1. Inspection is perfect in that the technician can always find the problem if there is one.2. Inspection and necessary replacements are performed at a regular interval of t = 28 days (1 month), regardless of the

item’s age, and all defective items are replaced.3. When a failure occurs, the item is replaced immediately from the existing stock (i.e. infinite supply).4. The time elapsed between the item being introduced (i.e. a new item) and the point u (when a defect is present) is

assumed to follow a Weibull distribution - this time length is called the initial time; the delay time distribution isassumed to be exponential (Baker and Wang, 1991).

5. There are N identical items to be inspected at the same inspection epoch.

The notation used for the purposes of our simulation and the subsequent presentation of the delay time based mainte-nance model (please refer to Section 4) is as follows:

xj
The number of replacements in the jth day since the start of simulation T Simulation length t The inspection interval (t = 28 days) for inspecting the item tr Previous replacement point of the item u The accumulated random initial time of the item, u = tr + u0 and u0 is the actual initial time h The random delay time of the item m The number of inspections before u, where mt < u < (m + 1)t Int(⁄) An integer function to return the largest integer less than or equal to argument ⁄
The relevant simulation routine is outlined in Fig. 5. The basic idea is to identify sequentially in which inspection intervalthe accumulated initial time is located, and then to check whether there is a failure or not depending on the length of thedelay time associated.

3.2. Age based inspection

According to an age based inspection scheme, inspection is scheduled in accordance with the age of the individual item,i.e. an item that has been just replaced will not be inspected again immediately. This scheme is commonly employed foritems positioned individually at different locations or for items of strategic importance, such as aircraft engines. In the casestudy described by Baker and Wang (1991), where the failure and inspection data of 105 volumetric and 35 peristaltic pumpsused in a large hospital was analysed, an age based inspection policy was used. The major failure mode of the pump was thefailure of the pressure transducer.

To stage the problem we either assume or use the practice adopted in the hospital.

1. Inspection is perfect in that the technician can always find the problem, if there is one (assumed).2. Inspection and replacement are undertaken at a regular interval of one month when defective transducers are replaced

(practice).3. When the technicians performed an inspection or replacement at failures they put a label on the pump showing the time

of the inspection or replacement so that the next inspection is taking place a month after that particular day (practice).4. The distribution for the time elapsed between the introduction of a new item and the point u is Weibull; the delay time

distribution is exponential (fitted result).5. The technicians goes around the hospital every 2 weeks to check the pumps but only those associated with a time elapsed

since the last inspection or replacement of more than 3 weeks are inspected again according to the label attached to thepump (practice). Note that this ensures an age based inspection policy according to which a pump that has just beeninspected or with a new transducer will not be inspected again at the next check up round.

The additional notation used in the simulation routine is as follows:

t
The interval (t = 14 days) for checking up the need for an inspection s The minimum inspection interval (3 weeks) tp The time of the last inspection
Here a check up means to check the label in order to make sure whether the pump needs an inspection or not; inspectionmeans an actual inspection after checking the label.

The simulation routine is outlined in Fig. 6. In the simulation, we use the fitted model parameters by Baker and Wang(1991) and make some further assumptions to initiate our simulation study.

i=1

Generate u and h

int( / )m u t=

Is mt T> ? Yes

No

Is ( 1) ?m t T+ >

Yes

No

A failure in (mt,T),

int( ) int( ) 1u h u hx x+ += +

rt u h= +

Yes No

Generate u′ and h′

ru t u′= + , h h′=

A PM replacement at (m+1)t,

( 1) ( 1) 1m t m tx x+ += +( 1)rt m t= +


ru t u′= + , h h′=

i=i+1

Is i>N?

No Yes

Is u h T+ > ?

Yes

No

A failure in (mt,(m+1)t),

int( ) int( ) 1u h u hx x+ += +

rt u h= +

Is ( 1)u h m t+ > + ?

Stop & print x

Fig. 5. Simulation routine for block based inspection.


The simulation is run over a 2000 time-unit length with the initial time and the delay time distributions are chosen asWeibull and Exponential respectively. The inspection interval is 28 time units so we have 71 inspection intervals over the2000 time-unit simulation length.

4. The delay time model of the probabilities of failures and inspection replacements

In this section, we formulate a delay time based maintenance model for the purpose of forecasting future replacements,i.e. the forthcoming requirements for spare parts.


Let U and H be the random variables of the initial and delay times of a random defect respectively, and fU(u), FU(u) andfH(h), FH(h) the pdf. and cdf. of u and h respectively, we then have:

fUðujU > ðk� 1Þt � trÞ ¼fU ððk�1Þt�trþuÞR1ðk�1Þt�tr

fU ðuÞdutr < ðk� 1Þt;0 < u � 1

fUðuÞ tr � ðk� 1Þt;0 < u <1

8<: : ð1Þ

i=1

Generate u and h

int( / )m u t=

Is mt T> ? Yes

No

Is ( 1) ?m t T+ >

Yes

No

Is ( 1) pm t t τ+ − > ? A failure in (mt,T),

int( ) int( ) 1u h u hx x+ += +

rt u h= +

Yes No


ru t u′= +h h′=

1m m= +

A PM replacement at (m+1)t,

( 1) ( 1) 1m t m tx x+ += +( 1)rt m t= +


ru t u′= +h h′=

i=i+1

Is i>N?

No Yes

Is u h T+ > ?

Yes

No

A failure in (mt,(m+1)t),

int( ) int( ) 1u h u hx x+ += +

rt u h= +

Is ( 1)u h m t+ > + ?

Yes*

No

Stop & print x

Fig. 6. Simulation routine for age based inspection, ⁄ another routine is needed to calculate tp .


Subsequently,

Prðthe ith item identified at kt to be faultyÞ ¼ PrðU < kt � tir jU > ðk� 1Þt � tir;H > kt � uÞ

¼Z t

0fUðujU > ðk� 1Þt � tirÞ

Z 1

kt�ufHðhÞdhdu

¼Z t

0fUðujU > ðk� 1Þt � tirÞð1� FHðkt � uÞÞdu; ð2Þ

where tir denotes the last renewal time of the ith item before kt. It follows that the expected number of inspection replace-ments at kt, denoted by E(Nskt), is

EðNsðktÞÞ ¼XN

i¼1

Prðthe ith item identified at kt to be faultyÞ: ð3Þ

For the expected number of failure based replacements in ((k � 1)t, kt) we need to make the following assumptions that ren-der the model simpler: (i) the probability of having another failure before the immediate next inspection epoch after thefailure has occurred is almost zero; (ii) the probability of finding the newly replaced item to be faulty again at the immediate


next inspection epoch is also zero. This is usually justified in terms of the inspection interval being much smaller than the lifespan. Based on these two assumptions it is obvious that we only need to model the probability of having a failure in((k � 1)t, kt), and it is given by:

Fig. 7.replace

Prðthe ith item having a failure inððk� 1Þt; ktÞ ¼ PrðU < kt � tr jU > ðk� 1Þt � tir;H < kt � uÞ

¼Z t

0

Z x

0fUðujU > ðk� 1Þt � tirÞfHðx� uÞdudx

¼Z t

0fUðujU > ðk� 1Þt � tirÞFHðt � uÞdu: ð4Þ

The expected number of failure based replacements in ((k � 1)t, kt), denoted by E(Nf((k � 1)t, kt)), is:

EðNf ððk� 1Þt; ktÞÞ ¼XN

i¼1

ðPrðthe ith item having a failure inððk� 1Þt; ktÞÞ þ o; ð5Þ

where o denotes the small missing part due to our assumption.Summing Eqs. (3) and (5) we then have that the expected number of replacements during ((k � 1)t, kt], denoted by

E(Nr((k � 1)t, kt)), is given by:

EðNrððk� 1Þt; kt�Þ ¼XN

i¼1

ðPrðthe ith item identified at kt to be faultyÞ

þ Prðthe ith item having a failure inððk� 1Þt; ktÞÞ þ o

¼XN

i¼1

Z t

0fUðujU > ðk� 1Þt � tirÞduþ o

¼XN

i¼1

FUðtjU > ðk� 1Þt � tirÞ þ o �XN

i¼1

FUðtjU > ðk� 1Þt � tirÞ; ð6Þ

Eq. (6) shows that E(Nr((k � 1)t, kt))is only governed by the conditional cdf. of u.


This model is more complicated than the one discussed above and here we follow the practice adopted in our age basedinspection simulation. At each check-up point for each item, we have two scenarios to consider,

Scenario (1): no inspection is needed at the current check-up point.Scenario (2): inspection at the current check-up point.

Both scenarios are illustrated in Fig. 7.We consider the interval from the current check-up point over the next two check-up periods (i.e. 28 time units). For sce-

nario (1), since the item was non-faulty at tp and is still working at tp + t, the initial time, U, must e larger than tp � tr and thedelay time, H must be larger than t. One of the three diagnostic events can occur over the interval (tp + t, tp + 3t): the item isfound to be non-faulty at tp + 2t revealed by the inspection; he item is found to be faulty at tp + 2t by inspection; and finally

No inspection Inspection Inspection

pt t+ 2pt t+

pt t+

pt…

Inspection No inspection

rt

ptrt…

(Scenario 2)

Current check-up time

pt t−

No inspection

Current check up point

3pt t+

2pt t+

(Scenario1)

Scenario (1) no inspection at tp + t and scenario (2) an inspection at tp, where tp is the time of an immediate previous inspection point and tr is the lastment point.


one failure occurs before tp + 3t. We assume again here that the probability of having more than one failure in (tp + t, tp + 3t)is zero and after a replacement at failure, the probability of U appearing after tp + 3t is almost one. This assumption serves thepurpose of mathematical simplicity but is also valid in practice since the probability of having another faulty item shortlyafter its installation is very small indeed. It follows:

Prð the ith item identified at tp þ 2t to be faultyÞ ¼ PrðU � tp þ 2tjU > tp � tir;H > tp þ 2t � ujH > tp þ t � uÞ

¼Z t

0fUðujU > tp � tirÞ

Z 1

2t�ufHðhjH > t � uÞdhdu

þZ 2t

tfUðujU > tp � tirÞ

Z 1

2t�ufHðhÞdhdu

¼Z t

0fUðujU > tp � tirÞð1� FHð2t � ujH > t � uÞÞdu

þZ 2t

tfUðujU > tp � tirÞð1� FHð2t � uÞÞdu: ð7Þ

The condition of H > t � u results from the fact that the item has not yet failed at time tp + t.

Prða failure inðtp þ t; tp þ 3tÞÞ ¼ PrðU < tp þ 3tjU > tp � tir;H < tp þ 3t � ujH > tp þ t � uÞ

¼Z t

0fUðujU > tp � tirÞFHð2t � ujH > t � uÞduþ

Z 2t

tfUðujU > tp � tirÞFHð2t � uÞdu

þZ 3t

2tfUðujU > tp � tirÞFHð3t � uÞdu: ð8Þ

For scenario (2) we also need to consider two possible events; namely, the item is identified to be faulty at tp + 2t or it failedbefore tp + 2t.

Prð the ith item identified at tp þ 2t to be faultyÞ ¼ PrðU � tp þ 2tjU > tp � tir;H > tp þ 2t � ujH > tp þ t � uÞ

¼Z 2t

tfUðujU > tp � tirÞð1� FHð2t � uÞÞdu; ð9Þ

and

Prðthe ith item having a failure inðtp; tp þ 2tÞÞ ¼ PrðU < tp þ tjU > tp � tir ;H < 2t � uÞ

¼Z 2t

0fUðujU > tp � tirÞFHð2t � uÞdu: ð10Þ

The expected number of replacements from the current check-up over the next 28 time units is:

EðNrðtp þ t; tp þ 2tÞÞ �XN

i¼1

Prðthe ith item identified at tp þ 2t to be faultyÞþPrðthe ith item having a failure inðtp þ t; tp þ 3tÞÞ

!dðiÞ

þXN

i¼1

Prðthe ith item identified at tp þ 2t to be faultyÞPrðtheith item having a failure inðtp; tp þ 2tÞÞ

!ð1� dðiÞÞ

¼XN

i¼1

Z 2t

0fUðujU > tp � tirÞduþ

Z 3t

2tfUðujU > tp � tirÞFHð3t � uÞdu

� �dðiÞ

þXN

i¼1

Z 2t

0fUðujU > tp � tirÞduð1� dðiÞÞ; ð11Þ

where

dðiÞ ¼1 if the ith item is in scenario ð1Þ0 otherwise

�:

Eqs. (5), (6), and (10) provide the forecasted demand for spare parts since these faulty or failed items must be replaced.For the remainder of the paper, the forecasted demand using the delay time maintenance model is termed as the ‘DT’ basedapproach.


5. Time series based forecasting model

We introduce the following notation:

t
The forecast revision period (t = 28 time units) zt Actual demand size at time t z0t Exponentially smoothed estimate of the demand size at time t updated only if demand occurs at time t pt Actual demand interval at time t p0t Exponentially smoothed estimate of the demand interval at time t, updated only if demand occurs at time t a Smoothing parameter ð0 � a � 1Þ; Ft Estimate of mean demand per period made at time t for period t +1.
The underlying model used for the purposes of the analysis presented in this section, represents the demand as a com-
pound process. Estimates are built from constituent elements, namely the demand size when demand occurs (zt, distributedwith a mean l and variance r2), and the inter-demand interval (pt). Both demand sizes and intervals are assumed to be sta-tionary; sizes and intervals are assumed to be independent. Demand is assumed to occur according to a Bernoulli process;subsequently, the inter-demand intervals are geometrically distributed (with mean p). There are no restrictions on the de-mand size distribution (Syntetos and Boylan, 2005). Under these conditions, the underlying mean level of the demand is: l/p.
According to Croston’s method, separate exponential smoothing estimates of the average size of the demand ðz0tÞ and theaverage interval between demand incidences ðp0tÞ are made after demand occurs (using the same smoothing constant value).If no demand occurs, the estimates remain exactly the same. The forecast, Ft for the next time period is given by:

Ft ¼z0tp0t

ð12Þ

where p0t ¼ p0t�1 þ aðpt � p0t�1Þ and z0t ¼ z0t�1 þ aðzt � z0t�1ÞThe method was claimed (Croston, 1972) to be unbiased and if demand occurs at every time period, Croston’s estimator is

identical to Single Exponential Smoothing (SES).The method is, intuitively at least, superior to SES and Simple Moving Average (SMA). Croston’s method is currently used

by leading statistical forecasting software packages and it has motivated a substantial amount of research work over theyears. Syntetos and Boylan (2001) showed that Croston’s estimator is biased. The bias arises since, if it is assumed that esti-mators of demand size and demand interval are independent, then:

Ez0tp0t

� �¼ Eðz0tÞE

1p0t

� �; ð13Þ

but

E1p0t

� �–

1Eðp0tÞ

; ð14Þ

and therefore Croston’s method is not unbiased. It is clear that this result does not depend on Croston’s assumptions of sta-tionarity and geometrically distributed demand intervals. The magnitude of the bias is estimated by Syntetos and Boylan(2005) where it is shown that the bias is approximately:

Bias � a2� a

l ðp� 1Þp2 : ð15Þ

Subsequently, Syntetos and Boylan (2005) proposed a new intermittent demand forecasting method, based on thisapproximation. The method was developed, based on Croston’s idea of building demand estimates from constituent ele-ments. It is approximately unbiased and has been shown to outperform Croston’s method on theoretically generated andempirical data. The new estimator of mean demand is as follows:

Ft ¼ 1� a2

� � z0tp0t; ð16Þ

where a is the smoothing constant value used for the intervals. In this paper, the same smoothing constant is used for de-mand sizes although, following the suggestion of Schultz (1987), a different smoothing constant may also be used.

The expected estimate of demand per unit time period as well as the variance of the estimates (sampling error of themean) produced by this method are given by (17) and (18) respectively.

EðFtÞ ¼ E 1� a2

� � z0tp0t

� �� l

p� a

2lp2 ; ð17Þ

VarðFtÞ ¼ Var 1� a2

� � z0tp0t

� �¼ 1� a

2

� �2Var

z0tp0t

� �; ð18Þ


where

Table 1Distribu

Weib

Vari

Varz0tp0t

� �� a

2� apðp� 1Þ

p4 l2 þ a2� a

a2� �

þ r2

p2

� �:

Eq. (16) is the equation we will use in this paper for forecasting purposes and is termed the ‘SBA’ based approach, (Syn-tetos–Boylan Approximation, SBA, Syntetos and Boylan, 2005).

6. Analysis of results


In this case we simulated a variety of scenarios to compare the forecasted demand produced by the DT based approachand the SBA approach presented earlier. First, we experimented with two scenarios with regards to the variability of u. This isbecause we believe that a better understanding of the defect arrival distribution will lead to a better estimation of the de-mand using the delay time based maintenance model. Subsequently, we also simulated different scenarios with regards tothe pattern of the demand data by varying the number of items in service and the parameters in the distributions of u and h.

We have considered only one-step-ahead forecasts (i.e. estimation of the demand over the next inspection interval). Thedata from the actual simulation was recorded as the failures per time unit and the number of inspection replacements ateach inspection point. Subsequently, we have aggregated that data to form the total demand within each inspection interval.To be able to initialise the SBA method, the data from the first 20 inspections was used as the within sample sub-set; fore-casts were then produced over the next 51 time periods (inspection intervals) by using both the SBA and DT approach. Since

tion parameters of the initial time (top: scale parameter, bottom: shape parameter).

ull distribution Mean initial time

200 400 600 800 1000

ance of initial time 6000 0.0045 0.0023 0.0016 0.0012 0.00092.7945 6.0042 9.2792 12.5721 15.8771

10,000 0.0044 0.0023 0.0016 0.0011 0.00092.1013 4.5422 7.0613 9.6031 12.1532

14,000 0.0044 0.0022 0.0015 0.0011 0.00091.7439 3.7722 5.8868 8.0273 10.1782

18,000 0.0045 0.0022 0.0015 0.0011 0.00091.5198 2.2807 5.1334 7.0412 8.9072

22,000 0.0045 0.0022 0.0015 0.0011 0.00091.3638 2.9336 4.5986 6.2941 8.0025

26,000 0.0046 0.0022 0.0015 0.0011 0.00091.2481 2.6723 4.1945 5.7487 7.3171

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

Time period

Dem

ands

by simulationby SBABy DT

Fig. 8. A simulation run and forecasted results using the SBA and DT approach.

6000 10000 14000 18000 22000 26000200

400

600

800

1000

Variance of the initial timeM

ean initial tim

e

20-30

10-20

0-10

-10-0

-5

0

5

10

15

20

25

30

6000 10000 14000 18000 22000 26000

Variance of the initial time

Diff

eren

ce b

etw

een

erro

rs

mean=200mean=400mean=600mena=800mean=1000

(a)

(b)

Fig. 9. (a) Contour plot of the differences between the average total absolute errors produced by SBA and DT, (b) The exact differences between the averagetotal absolute errors produced by SBA & DT (20 items and 20 simulations were used for each combination of the parameter values in Table 1).


the initial time distribution governs forecasting (as shown in Eq. (6)), we used a variety of the mean and variance of the ini-tial time to describe various situations. This was done due to the concern that we want to see in what situation that onemethod is better than the other. The parameters values under various means and variances of the Weibull based initial timedistribution are shown in Table 1.

The delay time is chosen as exponential with the scale parameter being 0.00174 after the analysis conducted in the casestudy performed by Baker and Wang (1991). Fig. 8 shows the results of a simulation run according to which there are 20items to be inspected and the scale and shape parameter for the initial time distribution are 0.00095 and 8.907 respectively.The forecasted values using both the SBA and DT approaches are presented with the dashed and dotted lines respectively.

Subsequently, we further consider the case of 20 items to be inspected, and we run 20 simulations for each combinationof the control parameter values in Table 1. We then compare the total absolute errors (over all 51 periods) resulting from theimplementation of the SBA and DT approach. The total absolute error is given by

Total absolute error ¼X51

i¼1

jforecasted demandi � actual demandij; ð19Þ

where i is the index for inspection interval i.The results in terms of the differences in average absolute errors between the SBA and DT are shown in Fig 9. For each

combination of the control parameter values in Table 1, the result was calculated in the following way

Difference ¼P

Total absolute errorsSBA

Number of simulations�P

Total absolute errorDT

Number of simulation

� �: ð20Þ

A positive number indicates that the SBA is worse than the DT and a negative number indicates the opposite. Sub-figure(a) corresponds to a contour plot with the x axis: the variances and y axis the means, and sub-figure (b) to the exact differ-ences between the average absolute errors produced by the two approaches. The colours in the contour plot indicate thelevel of the difference as shown in the legend.

Both figures show that the DT compares favourably to SBA; as the variance decreases and mean increases the advantageof the DT approach becomes even more obvious. The average absolute error is about 40, so the differences in some cases aresignificant.

Table 2Parameter values of the two pumps.

Volumetric pumps Peristaltic pumps

Initial time pdf Weibull – fUðuÞ ¼ aubuðauuÞbu�1e�ðauuÞbu au = 0.0017 au = 0.00073bu = 1.42 bu = 2.41

Delay time pdf Exponential – fHðhÞ ¼ ahe�ah h ah = 0.0174 ah = 0.009

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10Simulation runs

Abso

lute

erro

rs

By SBA

By DT

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10Simulation runs

Abso

lute

erro

rs

By SBABy DT

(a)

(b)

Fig. 10. (a) Volumetric pumps, average absolute errors, SBA = 87.63, DT = 83.29, (b) Peristaltic pumps, average absolute errors, SBA = 36.34, DT = 34.11.



In this case we use the parameter values for two pumps as discussed in the work conducted by Baker and Wang (1991).Following the practice outlined in the simulation section, we set the simulation length to 2000 time units and the forecastingperiod equal to 28 time units. The initial 20 periods are used for initiating the SBA method and the remaining 51 periods areused for forecasting and comparison purposes. Table 2 shows the parameter values for the two pumps. These values wereobtained by fitting various pdfs to the data and the best combination between the initial and delay time distributionsand parameters were chosen according to the AIC (Baker and Wang, 1991). There were 105 volumetric and 35 peristalticpumps. The main failure mode for the volumetric pumps was the transducer and the main failure mode of the peristalticpumps was the battery.

Fig. 10 shows the results.Fig. 10 confirms the earlier findings that the forecasts made by the DT approach are better than those resulting from the

application of the SBA method. From Table 2 we can see that the peristaltic pumps have a relatively long initial time. This, inconjunction with the relatively smaller number of pumps (35), as compared to the case of the volumetric pumps, leads to acomparatively higher degree of intermittence. However, it is interesting to note that in both cases, the DT based approachoutperformed the SBA method.

7. Conclusions and extensions

This paper presents a novel idea to forecast demand that relies upon the very sources of the demand generation processand then compare it with a well-known time series method. It is, we believe, the first of this type of study and makes a num-ber of significant contributions to the literature:


� First, it offers useful insights as to why the demand for spare parts is intermittent. There have been a number of consid-erable advancements in the area of intermittent demand forecasting in the recent years, all of which though have beenmainly focusing on coping, reactively, with the underlying structure of the demand patterns under concern. No attemptshave been made to characterise the very sources of such demand patterns for the purpose of developing more effective,pro-active, mitigation mechanisms. Such an approach would require taking a step back and looking at the industrialmaintenance processes that generate the relevant demand patterns. This is precisely what we do in this paper and in thatrespect we expect our work to initiate a new stream of research contributions in this area.� Second, we show how to use theory and models developed in maintenance research to facilitate the process of forecasting

intermittent demand. This is viewed as a very important issue both from a theoretical and practitioner perspective.� Third, we test the performance of our proposed approach in comparison with a time series method (Syntetos-Boylan

Approximation, SBA) using an experimental simulation-based framework that utilizes parameters and assumptions froman industrial case. Such experimental conditions offer also a practical relevance to our study.� Fourth, insights are offered into pertinent managerial issues along with a detailed discussion of the general conditions

under which one approach performs better than the other. Our analysis shows that if we can capture the failure and faultarrival mechanism of the items, maintenance based models should be used for forecasting purposes. If not or in very lowconfidence, SBA is a good alternative.

Demand intermittence occurs partly due to the fact that an item may have a long failure delay time, and therefore at thetime of inspection, more faulty items are identified and replaced at the same time. Maintenance and spare parts are closelyrelated with each other and should be treated as such. If the reliability characteristics of the items concerned can be cap-tured, then a maintenance driven spare parts model can be useful since it treats the problem differently from the conven-tional time series based forecasting methods. In particular, if the inspection scheme is to change in the future, then past datawill be of little use which may cause problems in time series based methods; however, the maintenance based model cancope with it. Along the same lines, if the item is newly commissioned then little past data will be available rendering theapplication of time-series methods problematic. However, if the design data is available or the manufacturer can providesome reliability data, then the maintenance based model can still be used.

The maintenance model considered for the purposes of our research is the delay time (DT) model. (The delay time conceptdefines a two stage-failure process: in the first stage the defect arises; the second stage covers the period from the momentthat the defect has aroused to failure.) Examples using simulated data confirm that the forecasts produced by the DT ap-proach are better than the SBA in almost all cases. The DT approach tends to perform even better when the variance andmean of the initial time are small and large respectively. We also used the fitted values of the distribution parameters ofthe hospital pumps used in a study conducted by Baker and Wang (1991) to test our model and that also led to the sameconclusion. In general, if the failure and fault arriving characteristics of the items can be captured, it is recommended touse a maintenance based model such as the delay time one discussed to forecast the spare part demand. If such informationis not available or the variance of the initial time (in the case discussed in this paper) is very large then the time series basedmethod should be used since it can also produce reasonable forecasts.

This paper compares only two exemplar forecasting methods to highlight the idea, and indeed there are many othertime series based forecasting methods and also maintenance models that could be used. However, we do have to saythat the models we used are very much representative of their respective approaches (although future studies shouldalso consider of course alternative methods). In particular the delay time maintenance model is perhaps the most rel-evant model to describe the inspection practice. One may question how can we capture the parameters in the delay timebased maintenance model in practical situations since unlike simulations, one will never know the exact value and for-mat of these parameters. Elaboration of this issue is beyond the scope of this paper and many previous studies haveproposed ways to select and estimate parameters (see, for example, Baker and Wang, 1991; Christer, 1999; Wang,2008a).

Finally, we acknowledge that there is certainly more to be done in this area and below we highlight some natural avenuesfor future research:

� The first is the relaxation of the assumption that the faulty item is replaced immediately upon an inspection which firstidentified the problem. If the delay time is relatively long (or in the case that spares are not available) then it may be pref-erable to delay the replacement until a more suitable time. This may render the use of a multi-stage delay time model anoption, where at least three stages should be used, e.g., normal, minor defect and serious defect. This offers the scope fordelayed replacement if the item is in the minor defect stage.� The second is to consider condition monitoring which is a further extension of the first with a possible continuous state

space. With condition monitoring we may identify the potential defect a lot earlier and then we may well delay thereplacement until the point that the normal spare part becomes available, if the delay time is long.� The third one is to consider repair as an option in addition to replacement if the item can be repaired. This assumes that

for certain damage the item may be repaired rather than replaced.� The fourth one is to include other PM activities such as lubricating, greasing, cleaning, and adjusting into the model,

which may alter the initial and delay time distributions.


� The fifth one is to consider the joint optimisation of production planning, maintenance and inventory since these threeare actually closely related. One can easily see that more production implies more equipment usage and degradationand therefore an increased requirement for spare parts. If the item is in a minor defect stage or identified by conditionmonitoring to be in the early stage of the defect, we may then lower the usage of the item until the needed spare partis available.� In addition, our proposed methodology offers a procedure to forecast the number of items that would be faulty in a time

span form the present check-up till the next inspection. However, and as one of the referees pointed out ,the lead time ofspare parts provision may often be greater than the inter-inspection period, in which case a stockist may wish of course tokeep in stock spare parts for possible failures that extend beyond the next inspection epoch. In our work we have beensolely concerned with forecasting related aspects and not with inventory control. If used in conjunction with inventorycontrol then forecasting the demand using our proposed method can be made for a number of inspection intervals ahead.� Further, and as discussed previously in this section, the comparisons performed in this paper may be extended in terms of

considering more time-series methods but also other maintenance models. Similarly, comparative performance may becaptured by a wide range of forecast accuracy measures, other than the one used in this study, such as the Relative Geo-metric Root Mean Squared Error that has been shown to be very robust in an intermittent demand context (Syntetos andBoylan, 2005).� The final extension of our work is to consider the combination of the time series based and maintenance based models to

produce robust forecasting results. More experiments are needed in this area.

Acknowledgements

The research described in this paper has been supported by the Engineering and Physical Sciences Research Council(EPSRC, UK) Grant Nos. EP/G006075/1, EP/G023042/1 and EP/F038526/1.

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spare pats demand 2011

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demand structures

lumpy demand

low demand size slow

intermittent demand

demand size distribution

interdemand intervals