Spare parts demand: Linking forecasting to equipment maintenance

Transportation Research Part E 47 (2011) 1194–1209

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Spare parts demand: Linking forecasting to equipment maintenance Wenbin Wang a,b, Aris A. Syntetos a,⇑ a b

Centre for Operational Research and Applied Statistics, University of Salford, UK 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 2010 Received in revised form 30 December 2010 Accepted 31 March 2011

Keywords: Spare parts Maintenance Forecasting Inventory management Delay time

a b s t r a c t Demand for spare parts is typically intermittent and forecasting the relevant requirements constitutes a very challenging exercise. Why is the demand for spare parts intermittent and how can we use models developed in maintenance research to forecast such demand? We attempt to answer these questions; we present a novel idea to forecast demand that relies upon the very sources of the demand generation process and we compare it with a wellknown time-series method. We conclude that maintenance driven models are associated with a better performance under certain conditions. We also outline an inter-disciplinary agenda for further research in this area. Ó 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 replacement. In some sectors, such as the aerospace and automotive industries, a wide range of service parts are held in stock, with significant 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 preventive maintenance is scheduled and is stochastic with regards to the demand size but deterministic as far as the demand arrival is concerned. Demand arising from corrective maintenance, after a failure has occurred, is stochastic with regards to the time arrival but deterministic in quantity (being one in most cases). Both require forecasting on the part of the stockist who is holding the relevant part. Such demand structures are typically intermittent in nature, meaning that demand arrives infrequently with (many) time periods showing no demand at all. In addition, demand when it occurs is not necessarily for a single unit, a very low demand size (slow demand) or a ‘constant’ requirement (clumped demand). That is to say, demand sizes may be highly variable leading to what is termed as ‘lumpy’ demand. In Fig. 1, the demand for two spare parts from the Royal Air Force (RAF, UK) is graphically depicted. In both cases demand is intermittent but is associated with differing profiles of the demand size distribution. Intermittent demand patterns are very difficult to deal with from a forecasting (and stock control) perspective because of the 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 been mainly focusing on coping, reactively, with the compound nature of the demand patterns under concern. However, no attempts have been made to characterise the very sources of such demand patterns for the purpose of developing more effec⇑ Corresponding author. Address: Centre for OR and Applied Statistics, Salford Business School, University of Salford, Maxwell Building, Salford M5 4WT, UK. Tel.: +44 161 295 5804. E-mail address: [email protected] (A.A. Syntetos). 1366-5545/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2011.04.008

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

1195

70

Demand (Units)

60 50 40 30 20 10 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Time period Slow demand

Lumpy demand

Fig. 1. Intermittent demand.

tive, pro-active, mitigation mechanisms. Such an approach would require taking a step back and looking at the industrial maintenance processes that generate the relevant demand patterns. This is precisely what we do in this paper. The following questions are addressed. Why is demand for spare parts intermittent? How can we use models developed in maintenance research 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 forecasting method 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 the research 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 natural extensions 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 on failures (corrective maintenance). For preventive maintenance, the timing is usually known in advance, but the demand for spare parts is stochastic. For such operations, part of the demand for spare parts may be known if previous condition monitoring or inspection revealed evidence of potentially defective items, or if an age or block based replacement policy is applied. However, in the majority of cases, inspections that are carried out may lead to further replacements. As such, demand due to preventive maintenance is still stochastic and therefore it needs to be forecasted. An illustrative example is that of Domestic cars’ service. Before a car arrives, the servicemen do not know which parts should be replaced. Unless a forecast is available and a safety stock decision has been made, new parts may need to be ordered. It may obviously take several days for those parts to arrive and this would extend the downtime and, consequently, the cost due to preventive maintenance. For unplanned repairs/replacements, the consequences of stockouts may include further production downtime with significant costs. Therefore, for both planned and unplanned maintenance, some kind of safety stock policy is required. Such a policy would 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 forecasting mean demand and not on the replenishment related aspects of the problem. A combined spare part inventory and maintenance model is to be addressed in the next steps of our research. Further consideration of the transportation related aspects of the problem is also an area that has not received sufficient attention in the literature. The integration of inventory management and transportation mode selection for spare parts logistics systems (e.g. Kutanoglu and Lohiya, 2008) constitutes 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’ requirements. 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 arrivals and demand sizes). In particular he suggested using Single Exponential Smoothing (SES) for separately forecasting the interval between demand incidences and the demand sizes, when demand occurs. The ratio of the latter over the former may then be used in order to estimate the mean demand per time period.

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Syntetos and Boylan (2001) showed that Croston’s estimator is biased and in a follow-up paper proposed a bias-adjusted method (Syntetos-Boylan Approximation, SBA, Syntetos and Boylan, 2005), in which Croston’s estimates are deflated by a factor of 1  a/2, where a is the smoothing constant used to update the SES estimates of the mean inter-arrival time for demands. The forecasts of mean demand are updated only if demand occurs; else they remain the same. Other adjustment factors 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 has been 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. As such, it constitutes the benchmark time-series approach (see Syntetos et al., 2009) to be used for forecasting demand in our study. 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 relies upon the reconstruction of the empirical distribution through a bootstrapping procedure. Such a procedure renders the estimation of the parameters of a hypothesised distribution redundant. The researchers claimed significant improvements in forecasting accuracy achieved by using their approach over Single Exponential Smoothing and Croston’s method. Nevertheless, further empirical evidence is required in order to develop our understanding of the benefits offered by such a nonparametric approach. Other bootstrapping methods for forecasting intermittent demands have also been discussed by Porras and 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 new results implemented into software products because of their practical importance (Fildes et al., 2008). Similar developments have 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 patterns. While practical implementations lag considerably behind theoretical propositions in this area, the relevant studies have 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 the underlying demand characteristics without questioning the demand generation process itself. Studying the demand generating process itself could help moving away from the re-active nature of current inventory management procedures for spare parts to more pro-active (in nature) mitigation mechanisms. Given the intuitive appeal of such an approach it is surprising that it has not been advanced in the academic literature. As discussed in the previous section the main research questions of this work are the following: ‘Why demand for spare parts is intermittent?’ and ‘How we may forecast the demand for spare parts based on a model developed in a maintenance context rather using time series approaches only?’ Before we attempt to answer 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 plant item concerned. In a typical maintenance setting two types of costs and downtimes are considered, namely, the downtime and cost due to corrective maintenance and the same due to preventive maintenance. Extensive efforts have been put into the optimisation of preventive maintenance intervals, replacement schedules and reliability improvements. See Cho and Parlar (1991), Dekker (1996), Nicolai and Dekker (2008), Thomas (1986), Wang (2002) for a series of review papers on this subject. It is noted however that most researches have treated maintenance as an area of research on its own, and did not consider the impact of the availability of spare parts on the plant downtime and cost due to maintenance. There are some exceptions; in particular for age based and block based maintenance policies (Scarf and Deara, 2003), inventory policies have been jointly considered with maintenance-related issues (Armstrong and Atkins, 1996; Brezavšcˇek and Hudoklin, 2003; de Smidt-Destombes et al., 2007, 2009; Kabir and Al-Olayan, 1994, Kabir and Al-Olayan, 1996). Nevertheless, the replacements resulting 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 parts provision (Albright and Gupta, 1993; Dhakar et al., 1994; Kim et al., 1996; Simpson, 1978; Yeralan et al., 1986). Attention has been paid to how equipment failures impact on the spare parts inventory policy. There are also a few review papers concerned 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 model and this is part of the focus of this paper. In particular, we seek a forecasting method that is based on regularly planned preventive maintenance activities as well as corrective maintenance activities using a modelling concept called delay time. The planned maintenance is not scheduled by an age or block based replacement policy; rather, we look at a general case where the need for spare parts is driven by the result of inspections. This is perhaps the most common scenario that one may encounter 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 an periodic inspection practice every t periods (weeks, months, . . .) with repair of failures undertaken as they arise. The inspection consists of a check list of activities to be undertaken and a general inspection of the operational state of the plant.

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h

0

u

time

s

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

3t

4t

time

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

Demand 2 1

t

2t

3t

4t

Time

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 minimise operational downtime. Other objectives could be considered, for example cost, availability or output. For now we focus on the 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 based studies 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 the period 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 a situation like the one depicted in Fig. 3. These items fail independently since they may be installed in different machines or vehicles. 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 factors influence 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 we may have more lumpy demand generated at the time of preventive maintenance. In the next section, we will generate a set of data using the above delay time modelling concept and then seek to forecast the demand by using the SBA statistical time series estimation procedure developed by Syntetos and Boylan (2005). The forecast results are to be compared to those developed 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 inspected at a fixed interval regardless of their age. This is typically the case when the items to be inspected constitute part of a larger system (Wang et al., 2010). Practical examples include production lines, gas turbines, offshore platforms and commercial vehicles. In our simulation study we assume that there are a number of identical components installed in the system. The failure process is simulated using the delay time concept discussed earlier. Both the time-series forecasting method and the delay time maintenance – based model to be shown later are used in order to forecast the demand. Forecasts are then

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to be compared to the actual demand data generated by the simulation. This will shed light on the extent to (circumstances under) which spare parts’ demand can be forecasted by appropriate maintenance models based on reliability characteristics and 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 is assumed 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 maintenance model (please refer to Section 4) is as follows: xj T t tr u h m Int(⁄)

The number of replacements in the jth day since the start of simulation Simulation length The inspection interval (t = 28 days) for inspecting the item Previous replacement point of the item The accumulated random initial time of the item, u = tr + u0 and u0 is the actual initial time The random delay time of the item The number of inspections before u, where mt < u < (m + 1)t 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 interval the accumulated initial time is located, and then to check whether there is a failure or not depending on the length of the delay 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 for items positioned individually at different locations or for items of strategic importance, such as aircraft engines. In the case study described by Baker and Wang (1991), where the failure and inspection data of 105 volumetric and 35 peristaltic pumps used in a large hospital was analysed, an age based inspection policy was used. The major failure mode of the pump was the failure 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 the pump (practice). Note that this ensures an age based inspection policy according to which a pump that has just been inspected 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

s tp

The interval (t = 14 days) for checking up the need for an inspection The minimum inspection interval (3 weeks) 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; inspection means 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.

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

1199

i=1 No Yes

Generate u and h

Is i>N? m = int(u / t )

Stop & print x

Yes

i=i+1

Is mt > T ? No Is (m + 1)t > T ?

No

Yes

Yes

Is u + h > T ? No

No

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

A failure in (mt,T),

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

xint( u + h ) = xint( u + h ) + 1

xint( u + h ) = xint( u + h ) + 1

tr = u + h

tr = u + h

Yes

A PM replacement at (m+1)t,

x( m +1) t = x( m +1) t + 1 tr = (m + 1)t

Generate u ′ and h′

Generate u ′ and h′

u = t r + u ′ , h = h′

u = t r + u ′ , h = h′

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 as Weibull and Exponential respectively. The inspection interval is 28 time units so we have 71 inspection intervals over the 2000 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. 4.1. Block based inspection Let U and H be the random variables of the initial and delay times of a random defect respectively, and fU(u), FU(u) and fH(h), FH(h) the pdf. and cdf. of u and h respectively, we then have:

fU ðujU > ðk  1Þt  t r Þ ¼

8 < RfU1ððk1Þttr þuÞ :

ðk1Þtt r

fU ðuÞ

fU ðuÞdu

tr < ðk  1Þt; 0 < u  1 : tr  ðk  1Þt; 0 < u < 1

ð1Þ

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

i=1 No Yes

Generate u and h

Is i>N? m = int(u / t )

Stop & print x i=i+1

Yes

Is mt > T ? No Is (m + 1)t > T ?

No

Yes Yes

Is u + h > T ? No

m = m +1

No

Yes

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

A failure in (mt,T),

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

xint( u + h ) = xint( u + h ) + 1

xint( u + h ) = xint( u + h ) + 1

tr = u + h

tr = u + h

Is ( m + 1)t − t p > τ ? Yes* No A PM replacement at (m+1)t,

x( m +1) t = x( m +1) t + 1 Generate u ′ and h′

tr = (m + 1)t

u = tr + u ′ h = h′

Generate u ′ and h′

u = tr + u ′ h = h′

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 Pr ðU < kt  tir jU > ðk  1Þt  t ir ; H > kt  uÞ Z 1 Z t fU ðujU > ðk  1Þt  t ir Þ fH ðhÞdhdu ¼ 0 ktu Z t fU ðujU > ðk  1Þt  t ir Þð1  F H ðkt  uÞÞdu; ¼

ð2Þ

0

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

EðNs ðktÞÞ ¼

N X

Prðthe ith item identified at kt to be faultyÞ:

ð3Þ

i¼1

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

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next inspection epoch is also zero. This is usually justified in terms of the inspection interval being much smaller than the life span. 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:

Prðthe ith item having a failure inððk  1Þt; ktÞ ¼ Pr ðU < kt  tr jU > ðk  1Þt  t ir ; H < kt  uÞ Z t Z x fU ðujU > ðk  1Þt  t ir ÞfH ðx  uÞdudx ¼ 0 0 Z t ¼ fU ðujU > ðk  1Þt  t ir ÞF H ðt  uÞdu:

ð4Þ

0

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 ¼

N X ðPrðthe ith item having a failure inððk  1Þt; ktÞÞ þ o;

ð5Þ

i¼1

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:

EðNr ððk  1Þt; ktÞ ¼

N X ðPrðthe ith item identified at kt to be faultyÞ i¼1

þ Prðthe ith item having a failure inððk  1Þt; ktÞÞ þ o N Z t X fU ðujU > ðk  1Þt  t ir Þdu þ o ¼ i¼1

¼

N X

0

F U ðtjU > ðk  1Þt  t ir Þ þ o 

N X

i¼1

F U ðtjU > ðk  1Þt  tir Þ;

ð6Þ

i¼1

Eq. (6) shows that E(Nr((k  1)t, kt))is only governed by the conditional cdf. of u. 4.2. Age based inspection This model is more complicated than the one discussed above and here we follow the practice adopted in our age based inspection 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 scenario (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 the delay time, H must be larger than t. One of the three diagnostic events can occur over the interval (tp + t, tp + 3t): the item is found 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

Current check up point Inspection

No inspection

Inspection

tp

tp + t

t p + 2t

… tr

(Scenario1)

t p + 3t

Current check-up time (Scenario 2)

No inspection Inspection … tr

tp − t

tp

No inspection

tp + t

t p + 2t

Fig. 7. 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 last replacement point.

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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 the purpose of mathematical simplicity but is also valid in practice since the probability of having another faulty item shortly after its installation is very small indeed. It follows:

Prð the ith item identified at tp þ 2t to be faultyÞ ¼ Pr ðU  t p þ 2tjU > t p  t ir ; H > t p þ 2t  ujH > t p þ t  uÞ Z t Z 1 ¼ fU ðujU > t p  tir Þ fH ðhjH > t  uÞdhdu 0

þ ¼

2tu

Z

2t

fU ðujU > tp  t ir Þ

t

Z

Z

1

fH ðhÞdhdu

2tu

t

fU ðujU > t p  tir Þð1  F H ð2t  ujH > t  uÞÞdu

0

þ

Z

2t

fU ðujU > tp  t ir Þð1  F H ð2t  uÞÞdu:

ð7Þ

t

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

Prða failure inðt p þ t; t p þ 3tÞÞ ¼ Pr ðU < t p þ 3tjU > t p  t ir ; H < t p þ 3t  ujH > t p þ t  uÞ Z 2t Z t fU ðujU > t p  tir ÞF H ð2t  ujH > t  uÞdu þ fU ðujU > t p  t ir ÞF H ð2t  uÞdu ¼ 0

þ

Z

t 3t

fU ðujU > tp  t ir ÞF H ð3t  uÞdu:

ð8Þ

2t

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

Prð the ith item identified at tp þ 2t to be faultyÞ ¼ Pr ðU  t p þ 2tjU > t p  t ir ; H > t p þ 2t  ujH > t p þ t  uÞ Z 2t fU ðujU > tp  tir Þð1  F H ð2t  uÞÞdu; ¼

ð9Þ

t

and

Prðthe ith item having a failure inðt p ; t p þ 2tÞÞ ¼ Pr ðU < t p þ tjU > tp  tir ; H < 2t  uÞ Z 2t ¼ fU ðujU > tp  tir ÞF H ð2t  uÞdu:

ð10Þ

0

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

EðNr ðt p þ t; t p þ 2tÞÞ 

N X

Prðthe ith item identified at t p þ 2t to be faultyÞ

!

þPrðthe ith item having a failure inðt p þ t; t p þ 3tÞÞ ! N X Prðthe ith item identified at t p þ 2t to be faultyÞ

dðiÞ

i¼1

ð1  dðiÞÞ Prðtheith item having a failure inðt p ; tp þ 2tÞÞ  Z 3t N Z 2t X fU ðujU > t p  t ir Þdu þ fU ðujU > tp  t ir ÞF H ð3t  uÞdu dðiÞ ¼ þ

i¼1

i¼1

þ

0

N Z X i¼1

2t 2t

fU ðujU > t p  t ir Þduð1  dðiÞÞ;

ð11Þ

0

where

 dðiÞ ¼

1 if the ith item is in scenario ð1Þ : 0 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’ based approach.

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5. Time series based forecasting model We introduce the following notation: t zt z0t pt p0t

The forecast revision period (t = 28 time units) Actual demand size at time t Exponentially smoothed estimate of the demand size at time t updated only if demand occurs at time t Actual demand interval at time t Exponentially smoothed estimate of the demand interval at time t, updated only if demand occurs at time t Smoothing parameter ð0  a  1Þ; Estimate of mean demand per period made at time t for period t +1.

a Ft

The underlying model used for the purposes of the analysis presented in this section, represents the demand as a compound process. Estimates are built from constituent elements, namely the demand size when demand occurs (zt, distributed with a mean l and variance r2), and the inter-demand interval (pt). Both demand sizes and intervals are assumed to be stationary; 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 demand 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 ðz0t Þ and the average interval between demand incidences ðp0t Þ 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 ¼

z0t p0t

ð12Þ

where p0t ¼ p0t1 þ aðpt  p0t1 Þ and z0t ¼ z0t1 þ aðzt  z0t1 Þ The 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 the years. Syntetos and Boylan (2001) showed that Croston’s estimator is biased. The bias arises since, if it is assumed that estimators of demand size and demand interval are independent, then:

 0   z 1 E t0 ¼ Eðz0t ÞE 0 ; pt pt

ð13Þ

  1 1 E 0 – ; pt Eðp0t Þ

ð14Þ

but

and therefore Croston’s method is not unbiased. It is clear that this result does not depend on Croston’s assumptions of stationarity 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 

a 2a

l

ðp  1Þ : p2

ð15Þ

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

 a z0t Ft ¼ 1  ; 2 p0t

ð16Þ

where a is the smoothing constant value used for the intervals. In this paper, the same smoothing constant is used for demand 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 the mean) produced by this method are given by (17) and (18) respectively.

EðF t Þ ¼ E

  a z0t l al   1 ; 2 p0t p 2 p2

VarðF t Þ ¼ Var

    0 a z0t a2 zt ¼ 1  ; 1 Var 2 p0t 2 p0t

ð17Þ

ð18Þ

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where

Var

 0   zt a pðp  1Þ  2 a 2  r2  : l þ a þ p4 p0t 2a 2a p2

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

6. Analysis of results 6.1. Block based inspection In this case we simulated a variety of scenarios to compare the forecasted demand produced by the DT based approach and the SBA approach presented earlier. First, we experimented with two scenarios with regards to the variability of u. This is because we believe that a better understanding of the defect arrival distribution will lead to a better estimation of the demand using the delay time based maintenance model. Subsequently, we also simulated different scenarios with regards to the 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). The data from the actual simulation was recorded as the failures per time unit and the number of inspection replacements at each 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; forecasts were then produced over the next 51 time periods (inspection intervals) by using both the SBA and DT approach. Since

Table 1 Distribution parameters of the initial time (top: scale parameter, bottom: shape parameter). Weibull distribution

6000 10,000 14,000 18,000 22,000 26,000

400

600

800

1000

0.0045 2.7945 0.0044 2.1013 0.0044 1.7439 0.0045 1.5198 0.0045 1.3638 0.0046 1.2481

0.0023 6.0042 0.0023 4.5422 0.0022 3.7722 0.0022 2.2807 0.0022 2.9336 0.0022 2.6723

0.0016 9.2792 0.0016 7.0613 0.0015 5.8868 0.0015 5.1334 0.0015 4.5986 0.0015 4.1945

0.0012 12.5721 0.0011 9.6031 0.0011 8.0273 0.0011 7.0412 0.0011 6.2941 0.0011 5.7487

0.0009 15.8771 0.0009 12.1532 0.0009 10.1782 0.0009 8.9072 0.0009 8.0025 0.0009 7.3171

3 by simulation by SBA By DT

2.5

2

Demands

Variance of initial time

Mean initial time 200

1.5

1

0.5

0 0

5

10

15

20

25

30

35

40

45

50

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

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

(a)

1205

1000

800

Mean initial time

600

400

20-30 10-20 0-10 -10-0

6000

200 10000 14000 18000 22000 26000

Variance of the initial time

Difference between errors

(b) 30 mean=200

25

mean=400 mean=600

20

mena=800 mean=1000

15 10 5 0 6000

10000

14000

18000

22000

26000

-5

Variance of the initial time 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 average total 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 initial time to describe various situations. This was done due to the concern that we want to see in what situation that one method is better than the other. The parameters values under various means and variances of the Weibull based initial time distribution 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 case study performed by Baker and Wang (1991). Fig. 8 shows the results of a simulation run according to which there are 20 items 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 combination of the control parameter values in Table 1. We then compare the total absolute errors (over all 51 periods) resulting from the implementation of the SBA and DT approach. The total absolute error is given by

Total absolute error ¼

51 X

jforecasted demandi  actual demandi j;

ð19Þ

i¼1

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

P Difference ¼

P  Total absolute errorsSBA Total absolute errorDT :  Number of simulations 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 differences between the average absolute errors produced by the two approaches. The colours in the contour plot indicate the level 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 advantage of the DT approach becomes even more obvious. The average absolute error is about 40, so the differences in some cases are significant.

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Table 2 Parameter values of the two pumps.

Initial time pdf Delay time pdf

Weibull – fU ðuÞ ¼ au bu ðau uÞ Exponential – fH ðhÞ ¼ ah e

bu 1 ðau uÞbu

e

Volumetric pumps

Peristaltic pumps

au = 0.0017

au = 0.00073

bu = 1.42

bu = 2.41 ah = 0.009

ah = 0.0174

ah h

Absolute errors

(a) 100 90

80 By SBA By DT

70

60 1

2

3

4

5

6

7

8

9

10

9

10

Simulation runs

(b)

45

Absolute errors

By SBA By DT

40

35

30

25 1

2

3

4

5

6

7

8

Simulation runs 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.

6.2. Age based inspection 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 forecasting period equal to 28 time units. The initial 20 periods are used for initiating the SBA method and the remaining 51 periods are used for forecasting and comparison purposes. Table 2 shows the parameter values for the two pumps. These values were obtained by fitting various pdfs to the data and the best combination between the initial and delay time distributions and parameters were chosen according to the AIC (Baker and Wang, 1991). There were 105 volumetric and 35 peristaltic pumps. The main failure mode for the volumetric pumps was the transducer and the main failure mode of the peristaltic pumps 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, in conjunction with the relatively smaller number of pumps (35), as compared to the case of the volumetric pumps, leads to a comparatively higher degree of intermittence. However, it is interesting to note that in both cases, the DT based approach outperformed 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 process and then compare it with a well-known time series method. It is, we believe, the first of this type of study and makes a number of significant contributions to the literature:

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 First, it offers useful insights as to why the demand for spare parts is intermittent. There have been a number of considerable advancements in the area of intermittent demand forecasting in the recent years, all of which though have been mainly focusing on coping, reactively, with the underlying structure of the demand patterns under concern. No attempts have 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 industrial maintenance processes that generate the relevant demand patterns. This is precisely what we do in this paper and in that respect 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 from an 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 fault arrival mechanism of the items, maintenance based models should be used for forecasting purposes. If not or in very low confidence, 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 the time of inspection, more faulty items are identified and replaced at the same time. Maintenance and spare parts are closely related with each other and should be treated as such. If the reliability characteristics of the items concerned can be captured, then a maintenance driven spare parts model can be useful since it treats the problem differently from the conventional time series based forecasting methods. In particular, if the inspection scheme is to change in the future, then past data will be of little use which may cause problems in time series based methods; however, the maintenance based model can cope with it. Along the same lines, if the item is newly commissioned then little past data will be available rendering the application of time-series methods problematic. However, if the design data is available or the manufacturer can provide some 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 concept defines a two stage-failure process: in the first stage the defect arises; the second stage covers the period from the moment that the defect has aroused to failure.) Examples using simulated data confirm that the forecasts produced by the DT approach are better than the SBA in almost all cases. The DT approach tends to perform even better when the variance and mean of the initial time are small and large respectively. We also used the fitted values of the distribution parameters of the hospital pumps used in a study conducted by Baker and Wang (1991) to test our model and that also led to the same conclusion. In general, if the failure and fault arriving characteristics of the items can be captured, it is recommended to use a maintenance based model such as the delay time one discussed to forecast the spare part demand. If such information is not available or the variance of the initial time (in the case discussed in this paper) is very large then the time series based method 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 other time series based forecasting methods and also maintenance models that could be used. However, we do have to say that the models we used are very much representative of their respective approaches (although future studies should also consider of course alternative methods). In particular the delay time maintenance model is perhaps the most relevant model to describe the inspection practice. One may question how can we capture the parameters in the delay time based maintenance model in practical situations since unlike simulations, one will never know the exact value and format of these parameters. Elaboration of this issue is beyond the scope of this paper and many previous studies have proposed 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 avenues for future research:  The first is the relaxation of the assumption that the faulty item is replaced immediately upon an inspection which first identified the problem. If the delay time is relatively long (or in the case that spares are not available) then it may be preferable to delay the replacement until a more suitable time. This may render the use of a multi-stage delay time model an option, where at least three stages should be used, e.g., normal, minor defect and serious defect. This offers the scope for delayed 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 the replacement 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.

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 The fifth one is to consider the joint optimisation of production planning, maintenance and inventory since these three are actually closely related. One can easily see that more production implies more equipment usage and degradation and therefore an increased requirement for spare parts. If the item is in a minor defect stage or identified by condition monitoring to be in the early stage of the defect, we may then lower the usage of the item until the needed spare part is 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 of spare parts provision may often be greater than the inter-inspection period, in which case a stockist may wish of course to keep in stock spare parts for possible failures that extend beyond the next inspection epoch. In our work we have been solely concerned with forecasting related aspects and not with inventory control. If used in conjunction with inventory control 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 be captured by a wide range of forecast accuracy measures, other than the one used in this study, such as the Relative Geometric Root Mean Squared Error that has been shown to be very robust in an intermittent demand context (Syntetos and Boylan, 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.

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