Content uploaded by Robert Boute

Author content

All content in this area was uploaded by Robert Boute on Apr 01, 2019

Content may be subject to copyright.

Forecasting Spare Part Demand using Service Maintenance Information

Sarah Van der Auweraera, Robert N. Boutea,b

aResearch center for Operations Management, Faculty of Business and Economics KU Leuven, Belgium

bTechnology & Operations Management Area, Vlerick Business School, Belgium

Abstract

We focus on the inventory management of critical spare parts that are used for service maintenance. These

parts are commonly characterized by a large variety, an intermittent demand pattern and oftentimes a high

shortage cost. Specialized service parts models focus on improving the availability of parts whilst limiting

the investment in inventories. We develop a method to forecast the demand of these spare parts by linking it

to the service maintenance policy. The demand of these parts originates from the maintenance activities that

require their use, and is thus related to the number of machines in the ﬁeld that make use of this part (known

as the active installed base), in combination with the part’s failure behaviour and the maintenance plan.

We use this information to predict future demand. By tracking the active installed base and estimating the

part failure behaviour, we provide a forecast of the distribution of the future spare parts demand during the

upcoming lead time. This forecast is in turn used to manage inventories using a base-stock policy. Through

a simulation experiment, we show that our method has the potential to improve the inventory-service trade-

oﬀ, i.e., it can achieve a certain cycle service level with lower inventory levels compared to the traditional

forecasting techniques for intermittent spare part demand. The magnitude of the improvement increases for

spare parts that have a large installed base and for parts with longer replenishment lead times.

Keywords: Inventory management, Forecasting, Spare parts, Service maintenance, Installed base

1. Introduction

In this paper we propose a forecast method for spare part demand by making use of information on

the maintenance operations. We address the following research questions: (1) How can we use information

from maintenance operations to generate forecasts, and (2) to what extent does this method improve the

inventory management of the spare parts, compared to more traditional forecast methods that only rely on

historical part demand to construct forecasts.

Eﬃcient spare part management is indispensable for companies that increasingly oﬀer services related

to the products they originally supply. Indeed, the after-sales activities (such as maintenance and repairs)

generate a more steady stream of revenue and can as such be an important source of proﬁt (Auramo &

Accepted for publication by the International Journal of Production Economics (March 2019)

Ala-risku, 2005; Cohen et al., 2006). At the same time, challenges are associated with managing these

after-sales activities in a cost-eﬀective way. The demand for spare parts, for instance, is very hard to

predict. When a critical part is requested and not available in stock, the company is not able to perform

the maintenance operation in time. This could jeopardize a client’s productivity, causing time delays and

high costs. To prevent part unavailability, companies keep stock buﬀers to deal with the uncertainty in part

demand. Keeping extra stock, however, can become costly when this is done for a large number of items, or

for expensive items. Eﬃcient spare parts inventory management is thus an important but challenging task,

where the investment in inventories is to be traded oﬀ against the desired service levels.

1.1. Background and Motivation

Our research is inspired by an Original Equipment Manufacturer (OEM) in the compressed air, generator,

and pump industry who is currently facing these challenges. The OEM has service maintenance contracts

with its customers, in which they guarantee a pre-deﬁned availability of spare parts. The OEM is thus

responsible for the maintenance of the machines installed at its customers. We refer to those assets as the

active installed base, i.e., the number of sold (installed) machines that haven’t been discarded, and can

generate spare parts demand either to replace failed components or for preventive maintenance. The OEM

currently manages over 250 000 diﬀerent spare parts. The demand for approximately 40 000 of these parts

is currently forecasted using simple exponential smoothing (SES) and inventories are managed using a base-

stock policy. For the remaining 210 000 parts demand is too small (only one or two items per year) or

too little historical demand observations are available to keep the part in inventory. The company aims to

increase the part availability in the future, without unduly inventory investments. To date, the spare part

inventory is managed by the Service Logistics department. The Service Operations department, however,

who is in charge of the maintenance operations itself, is increasingly collecting data on its installed base, such

as machine survival and part reliability data. The Service Logistics department is therefore investigating

whether they can make use of this information to improve their spare part inventory management.

Managing spare part inventory is challenging, as many of them are characterised by an intermittent

demand pattern: many periods of zero demand interspersed by occasional non-zero demands (e.g., Boylan

& Syntetos, 2010). As standard forecasting techniques produce inaccurate results for these intermittent

demands, speciﬁc techniques have been developed, with Croston-based methods the most widely used (Cros-

ton, 1972; Syntetos et al., 2009a; Boylan & Syntetos, 2010; Bacchetti & Saccani, 2012). These techniques

are time-series based in the sense that rely on the assumption that future values of the demand series can

be predicted based on past demand, and no attempt is made to identify variables that inﬂuence the series

(Stevenson, 2012). We believe that we can use more detailed sources of information, such as reliability

2

characteristics and the maintenance policy, to generate forecasts, rather than merely looking at past demand

data. The amount of data collected and stored by companies increases, which makes the use of more sophis-

ticated data driven methods possible. These methods allow to capture the drivers of the demand, instead of

the observed demand alone.

1.2. Related Literature

There are a number of papers that discuss the use of installed base information for spare part demand

forecasting (see Van der Auweraer et al., 2019). The demand drivers that are most commonly discussed in

literature are the maintenance policy, which depicts when a part is replaced (e.g., Hu et al., 2015; Romeijnders

et al., 2012), the size and age of the installed base (e.g., Jalil et al., 2011; Kim et al., 2017; Stormi et al.,

2018), and the part failure probability (e.g., Barabadi, 2012; Si et al., 2017; Ritchie & Wilcox, 1977). Those

works tend to focus on only a speciﬁc subset of installed base information and the combination of these

diﬀerent demand drivers is generally lacking. Moreover, the literature typically makes a distinction between

forecasting models for spare part demand for preventive maintenance (e.g., Hu et al., 2015; Romeijnders

et al., 2012; Wang & Syntetos, 2011; Wang, 2011) and forecasting demand for corrective maintenance (e.g.,

Hong et al., 2008; Jin & Liao, 2009; Kim et al., 2017; Minner, 2011; Stormi et al., 2018), sometimes also

referred to as reliability based forecasting. Both are rarely combined, however.

The literature on reliability based forecasting tends to focus on a speciﬁc phase in the product life cycle

(PLC) of the installed base, oftentimes the initial (e.g., Jin & Liao, 2009) or end-of-life (EOL) phase (e.g.,

Hong et al., 2008; Kim et al., 2017). By restricting the analysis to a speciﬁc PLC phase, simplifying model

assumptions can be made. For example, when the end-of-life phase is studied, potential expansion of the

installed base in the future can be neglected, which reduces the complexity of the forecasting problem.

However, these models are not applicable in the other life cycle phases and are in this way limited. In

addition, the majority of the methods makes restrictive distributional assumptions, for example on the

failure behaviour of the parts. Exceptions to this are Minner (2011) and the extension of his model by

Wu & Bian (2015). Their methods are applicable in every phase of the PLC and are not restrictive on

the distributional assumptions. Nonetheless, these approaches rapidly become complex and computationally

intensive because of their recursive nature, which renders them hard to apply in a practical setting.

When it comes to the beneﬁt of using information from the service operations, we ﬁnd that the proposed

methods in the literature generally show promising performance: they are capable to predict the actual

demand well and increase forecast accuracy. Nevertheless, their performance evaluation is typically decoupled

from inventory management and their impact on inventories and availability is rarely evaluated. This is in

contradiction to Syntetos et al. (2010), who state that forecasting performance in an inventory context should

3

always be evaluated with respect to its implications for stock control.

1.3. Our Contribution to the Literature

Our work contributes to the existing literature by oﬀering a versatile forecast method that considers

a larger set of spare part demand drivers (in combination with each other) to construct the forecasts. It

incorporates information on the maintenance policy, which can be either corrective or preventive. Corrective

maintenance generates stochastic demand, whereas preventive maintenance interventions generate deter-

ministic demand, as they are known in advance. Our method also takes into account the evolution of the

active installed base over time, which can increase through the sales of new machines that are serviced and

decrease through the end-of-use of old machines. Finally, our method considers the part reliability, which

deﬁnes when a part in the installed base will fail and require a corrective replacement. We refer to this

collection of information used as Service Maintenance Information (SMI).

Our method is applicable throughout the full PLC, and is not tailored to one speciﬁc phase. Consequently,

there is no need for separate methods for the diﬀerent PLC phases, and there is no need to identify the current

PLC stage. Moreover, it does not put restrictions on the probability distributions of the diﬀerent model

components as it uses real data observations to generate its forecast. The method uses a statistical ﬁt on the

historical part failures, and continuously improves this ﬁt (“learns”) as more information becomes available.

The versatility of our method makes it applicable for a large variety of parts, regardless of the part speciﬁc

characteristics and without any distributional assumption.

We also consider the data requirements for our method to be amenable for implementation in a practical

setting. Our method requires data of historical machine sales and discards to track the evolution of the

installed base over time, a history of past part failures to estimate the part failure probability, and information

on past preventive maintenance actions on the installed machines together with the current preventive

maintenance schedule –if applicable.

The result of our forecast is an estimate of the distribution of the demand during the upcoming lead

time. This distribution can be used as an input for an inventory policy; we apply it to determine the optimal

base-stock policy to attain a pre-deﬁned target service level. This way, we connect forecasting directly with

inventory management.

We ﬁnd that our method has the potential to improve forecast accuracy and outperform the current

state-of-the-art forecasting procedures for intermittent demand. The proposed method is able to achieve a

higher inventory-service eﬃciency, especially when the installed base is large and lead times are long.

The paper is organised as follows: In Section 2 we introduce and describe our forecasting method. We

evaluate its performance in a numerical study, which we introduce in Section 3. The results of this numerical

4

tThe current point in time. t >0.

LThe replenishment lead time.

jThe machine age; t– time of machine installation. j≤t.

iThe part age; t– time of part installation. i≤j.

pi,j,t,L The probability of a demand occurring for a part of age i, installed in a machine of age j,

in the time interval (t, t +L].

pm

j,t,L The conditional probability of a machine with age jto survive after the lead time L, esti-

mated at time t.

pp

i,t,L The conditional failure probability of a part with age iin (t, t +L], estimated at t.

nij (t) The counted number of parts with age i, installed in machines with age j, at t.

b

Dt,L The demand forecast made at time tover lead time L.

NtThe size of the active installed base at time t.

b

λt+sThe estimated number of future machine sales occurring in period t+s, estimated at t.

s∈ {1, ..., L}.

TpThe lifetime of a part (time from replacement until failure)

TmThe economic lifetime of a machine (time from installation until discard)

fTp(θ) The probability distribution of the time until part failure.

fTm(θ) The probability distribution of the time until machine discard.

Table 1: Notations used.

study are discussed in Section 4. Section 5 concludes the paper.

2. Installed Base Forecasting for Future Service Part Demand

In this section we present our approach to use information on the maintenance operations to forecast

future spare parts demand. We distinguish between three main sources of information: (1) the maintenance

policy, (2) the size and age of the installed base, where we distinguish between the installation of the machine

and the installation of the part, and (3) the part reliability. These model components are explained in further

detail in the following sections. Table 1 presents the notation used.

The spare part demand forecast, b

Dt,L, is made at time tfor the upcoming replenishment lead time L.

This timeframe is denoted by the discrete time interval (t, t+L]. The lead time Lis considered deterministic,

and in our company example its value can range from a week to over ﬁve months. At time t, we estimate

the probability bpi,j,t,L that a part of age iinstalled in a machine with age jgenerates a demand in the

upcoming lead time L. Next, we monitor how many parts nij(t) of age iare installed in machines with

age j. The forecast b

Dt,L then consists of the sum of the demand probabilities for each of the parts in the

installed base. In the remainder of this section we ﬁrst describe how the demand forecast is derived when

a corrective maintenance policy is in place (i.e., with only corrective replacements). Thereafter, in Section

2.6 we discuss the demand forecast in case a preventive maintenance policy applies (with corrective and

preventive replacements).

5

2.1. The Active Installed Base

The active installed base can be deﬁned as the set of systems or products for which a company provides

after-sales services (Dekker et al., 2013). It is the active set of machines, which contain the part under study

and can generate a future demand for spare parts. This is the number of machines that has been sold to

customers and has not been discarded yet. A machine discard occurs when a customer ends the use of a

machine. We observe the active installed base at time tand note the age jof each active machine. The age

jof the machine is a discrete number and represents its time since installation.

Machines typically last longer than their parts, which are replaced upon failure of the part (or preven-

tively). Therefore, we also monitor the age iof the studied part in each installation, i.e., the time since the

last part replacement, or if the part has not been replaced yet, the time since installation. We explicitly

distinguish between part age and machine age, contrary to, for example, Kim et al. (2017), because a demand

occurs for a machine that is still active and experiences a failure, and machines can be discarded from the

active installed base before they have failed, which means they can no longer generate a spare part demand.

The part age is a discrete number, expressed in the same time unit as the machine age. Thus, the part age

is always smaller than or equal to the age of the machine in which it is installed: i≤j. The part age idrives

the part failure probability, and the machine age jdrives the machine survival probability.

Let Ntdenote the size of the active installed base at time t, which corresponds to the sum of the number

of parts nij (t) of age iinstalled in machines of age j, for each part-machine age combination at time t:

Nt=

t

X

j=1

j

X

i=1

nij (t).(1)

As the installed base is monitored at each time t, we do not need to make assumptions on its evolution

over time, and its product life cycle behaviour is automatically captured.

In practical settings, monitoring the installed base implies keeping track of the machine sales and discards

to determine the size and age of the installed base, and of the maintenance interventions, i.e., the timing

of the corrective and preventive part replacements. As such, the company can determine nij (t) by simply

counting the number of parts of age i, installed in active machines of age j.

2.2. Demand Probability

Every part of age i, installed in a machine of age j, has a probability of being replaced in the upcoming

lead time, that is in the interval (t, t +L]. This probability is denoted by pi,j,t,L, which depends on both

the failure probability of the part, deﬁned by the part age i, and the probability that the machine is not

discarded during the lead time, depending on the machine age j. The demand probability pi,j,t,L is therefore

deﬁned by the product of (1) the failure probability of the part with age iin the interval (t, t +L], pp

i,t,L,

6

which is independent of the machine age j, and (2) the survival probability of a machine with age j,pm

j,t,L,

which is independent of the part age iand the part failure behaviour. Let Tpdenote the lifetime of a part

(time from replacement until failure) and Tmthe economic lifetime of a machine (from installation until

discard). We can then express pi,j,t,L as:

pi,j,t,L =P(Tp≤i+L|Tp> i)·P(Tm> j +L|Tm> j)

=pp

i,t,L ·pm

j,t,L.

(2)

We estimate the probability of pi,j,t,L using an estimation of the part failure probability pp

i,t,L, and an

estimation of the machine survival probability pm

j,t,L, which are discussed next.

2.3. Machine Survival Probability

The machine lifetime Tmis stochastic, but we assume that machines installed at diﬀerent points in time

have an identical lifetime probability distribution fTm. We consider machine lifetime independent of the

part lifetime. This implies that the lifetime of a machine solely depends on the machine age j, and not on

the age of the consisting parts nor the number of historical failures this machine has already experienced.

This independence seems a reasonable assumption if the machines are expensive relative to the parts (mix

of expensive and cheaper items). In that case, a machine will not be discarded if it experiences part failures,

but it will be repaired and parts will be replaced. This is in contrast to, for example, consumer electronics,

where the decision to discard the product is highly dependent on the number of historical failures of the

product and the customer’s willingness to repair (again) (Lu & Wang, 2015).

The probability at time tthat the machine is not discarded in the time interval (t, t +L] is deﬁned by

pm

j,t,L =R∞

j+LfTm(θ)dθ

R∞

jfTm(θ)dθ .(3)

Typically, fTm(θ) is not known by the service provider. However, when a company keeps track of the

evolution of its installed base over time, it can estimate fTm(θ), and consequently pm

j,t,L, using censored

observation data of the historical machine lifetime data.

2.4. Part Failure Behaviour

Without loss of generality, we assume there is no inﬂuence of diﬀerent usages or the operating environment

on the failure probability distribution, and parts installed at diﬀerent points in time have an identical

lifetime probability distribution fTp(θ). We consider parts that are non-repairable, i.e., need replacement

and generate a part demand upon failure. When replaced, we assume the part to be as-good-as-new.

Analogously to the probability of a machine discard, we determine for each of the parts in the active

installed base the conditional probability of failure during the upcoming lead time pp

i,t,L, given that the part

7

did not fail yet, and has reached age iat time t:

pp

i,t,L =Ri+L

ifTp(θ)dθ

1−Ri

0fTp(θ)dθ ,(4)

where fTp(θ) is the probability distribution of the time-to-failure.

In some cases, fTp(θ) can be estimated directly using data from, for example, stress testing. An alternative

would be to keep track of historical corrective and preventive replacements, in which case the company needs

to register the type of replacement with the associated part ages upon replacement, as well as the age of parts

which are still operational at the time of observation. Those part ages are in fact the part survival time.

For failed parts, the survival time is recorded as a non-censored observation. For preventively replaced parts

and parts which are still operational, this survival time is a lower bound; it is registered as a right-censored

observation. Moreover, for parts that did not fail but whose machine is no longer active, the part survival

time gets a lower bound as well. This way, a distribution can be ﬁt and pp

i,t,L can be estimated. Because the

machine discards are implicitly taken into account in the estimate, pm

j,t,L is already included in the estimation

of the part failure probability pp

i,t,L.

2.5. Lead Time Demand Forecast

To obtain the lead time demand forecast we ﬁrst estimate the demand probabilities for the diﬀerent part

age - machine age combinations in the installed base using an estimator of the part failure probability and

an estimator of the machine surivial probability:

bpi,j,t,L =bpp

i,t,L ·bpm

j,t,L,(5)

which we then multiply with the respective ni,j (t) to obtain the demand estimated at time tover the risk

period1L. The lead time demand forecast b

Dt,L then equals:

b

Dt,L =

t

X

j=1

j

X

i=1

nij (t)·bpi,j,t,L.(6)

In addition, we can also include the possibility of new machines sales that cause a part failure during

that same lead time. We acknowledge that in most cases the eﬀect will be limited because the duration of

the lead time is typically smaller compared to the time to failure. The number of new installations during

the lead time can be estimated either by assuming a sales process, or by including projections from the

marketing department. The expected number of sales in period t+s(s∈ {1, ..., L}) is denoted by b

λt+s.

1Note that in a periodic review inventory policy, demand is forecasted over the risk period, which is deﬁned by the lead time

plus the review period.

8

Each of the newly installed items has an associated demand probability of pi,j,t+s,L−s, where i=j= 0 as

both the part and machine have age zero upon installation. The forecasted part demand, estimated at time

tover L, is then given by:

b

Dt,L =

t

X

j=1

j

X

i=1

nij (t)·bpi,j,t,L +

L

X

s=1 b

λt+s·bp0,0,t+s,L−s.(7)

In order to use the forecast as a decision input for an inventory policy we need to forecast the distribution

of the lead time demand. For each part in the installed base, the probability of demand is expressed by

bpi,j,t,L. The forecast of the probability distribution of the lead time demand at time t,b

P(Dt,L) is then

deﬁned by a Poisson binomial distribution, i.e., a sum of independent Bernoulli random variables which may

have diﬀerent expectations, namely bpi,j,t,L (Hong, 2013). Note that when the expectations, bpi,j,t,L, are equal

for all iand j, the Poisson binomial distribution boils down to a binomial distribution.

With an active installed base of size Ntat time tand the forecasted lead time demand probability at

time tfor part k, denoted as bpk(i.e., the corresponding pi,j,t,L for part k), k∈[1, Nt], we then have the

following forecast of the lead time demand probabilities:

b

P(Dt,L = 0) =

Nt

Y

a=1

(1 −bpa),

...,

b

P(Dt,L =dt,L) = X

A∈Zn Y

a∈Abpa·Y

b∈Ac

(1 −bpb)!,

...,

b

P(Dt,L =Nt) =

Nt

Y

a=1 bpa,

(8)

with Znthe combinations of parts in the active installed base that can be selected from {1 ,2, . . . , Nt}for

which the total demand equals dt,L. For example, if we estimate the probability of demand to be equal to

two ( b

P(Dt,L= 2)), with an active installed base consisting of three parts (Nt= 3), then Znconstitutes all

the combinations of two part numbers that can generate a demand of two: Zn={(1,2),(2,3),(1,3)}.Ais

then the set of possible parts that eﬀectively generate a demand (i.e., fail) in the subset of Zn, for example

A={1,2}, and Acis its complement: Ac={1 ,2, . . . , Nt} \A, in this case Ac= 3. In our example, we

would then obtain b

P(Dt,L = 2) = (bp1·bp2)·(1 −bp3)+(bp2·bp3)·(1 −bp1)+(bp1·bp3)·(1 −bp2).

2.6. Preventive Maintenance Policy

Spare part demand not only originates from unscheduled corrective replacements (due to failures), but

also from preventive replacements. The latter stems from scheduled maintenance, aimed at preventing

9

Figure 1: Illustration of preventive replacements, with τ= 8 and Nt= 2.

failures by replacing parts before they fail. We assume that for both corrective and preventive maintenance

actions, spare parts replace the failed or worn parts.

Preventive maintenance can be time-based, i.e., a part is replaced every so many running hours or after

a certain calendar time, regardless of its condition, or inspection-based, where a part is inspected after a

certain time and it is decided upon whether to replace the part or not. Between two preventive replacements,

a breakdown might still occur, calling for a corrective replacement. When a time-based periodic replacement

policy is in place and the planned periodic maintenance interventions are not rescheduled due to intermediate

corrective interventions within the periodic maintenance interval, the preventive replacements generate a

deterministic demand stream. When such a preventive policy applies, we can exploit the knowledge about

planned maintenance, which is known as advance demand information, to predict its demand.

We denote τas the periodic maintenance interval, i.e., the calendar time between two preventive main-

tenance interventions. A larger interval τreduces the total number of preventive replacements but increases

the probability of an unplanned corrective intervention (if τapproaches ∞, the preventive policy reduces to

a corrective policy). A short interval τ, however, does not fully utilise the part lifetime. The optimal length

of the maintenance interval τis the result of this tradeoﬀ (e.g., Jardine & Tsang, 2013). In our work, we

consider the value of τas exogenously given.

The actual planning of the time-based periodic part replacement is based on the time since installation.

We deﬁne tj,P M as the time of the next periodic replacement of the part under study for a machine with

age j. As such, tj,P M equals the time of installation of the machine, (t−j) , plus the smallest value that

is a multiple of the periodic maintenance interval and exceeding the machine age j. If we deﬁne Tas the

set {τ, 2τ, 3τ,. . . }, then tj,P M = (t−j) + min{x|x∈ T , x > j}. This is illustrated in Figure 1, where a

demand forecast is made at time 10, the installed base consists of two machines, installed at time 1 and 7

respectively, with τ= 8. At time t= 10, the age of the ﬁrst installed machine equals 9 and the age of the

second installed machine equals 3. As such, t9,P M = (10 −9) + 2 ×8 = 17 and t3,P M = (10 −3) + 1 ×8 = 15.

If the lead time L= 7, the forecast for the preventive maintenance demand (during the lead time) at time

t= 10 equals 2.

10

The total number of replacements in the upcoming lead time Dt,L is at least as large as the number of

planned periodic replacements, that is, larger than or equal to:

Nt

X

k=1

Ik,tj,P M (9)

where Ik,tj,P M is an indicator function that equals one if for part k, installed in a machine with age j,

tj,P M ∈(t, t +L], and zero otherwise. To account for discards we can correct Equation (9) with the number

of planned discards in (t, t +L].

When a preventive maintenance policy is in place, the demand originating from corrective replacements

during the lead time is forecasted by the probability that a part with age ifails during the lead time,

but prior to the periodic replacement, that is in the interval (t, min(tj,P M , t +L)]. Let πbe the age of

the part at the time of the periodic replacement or at the end of the lead time, whichever comes ﬁrst:

π= min{i+ (tj,P M −t), i +L}. Then:

pp

i,j,t,L =Rπ

ifTp(θ)dθ

1−Ri

0fTp(θ)dθ ,(10)

where fp(Tp) is the probability distribution of the time-to-failure, estimated based on censored failure ob-

servations. Remark that under a preventive policy the failure probability pp

i,j,t,L becomes dependent on the

machine age, as it determines the timing of the periodic replacement.

The forecasted distribution of the total part demand for both preventive and corrective replacements in

the upcoming lead time is then given by:

b

P(Dt,L =

Nt

X

k=1

Ik,tj,P M + 0) =

Nt

Y

a=1

(1 −bpa),

...,

b

P(Dt,L =

Nt

X

k=1

Ik,tj,P M +dt,L) = X

A∈Zn

(Y

a∈AbpaY

b∈Ac

(1 −bpb)),

...,

b

P(Dt,L =

Nt

X

k=1

Ik,tj,P M +Nt) =

Nt

Y

a=1 bpa.

(11)

3. Numerical Study

We set up a simulation experiment to illustrate the beneﬁts of our forecasting method using Service

Maintenance Information (SMI). The prime benchmark method in our study is the Syntetos-Boylan Ap-

proximation (SBA) method discussed by Syntetos & Boylan (2005). Hasni et al. (2018) show in a recent

11

study the potential outperformance of this method compared with bootstrapping techniques for intermittent

demand, and as such advocate the value of the SBA method as a benchmark. In addition, we compare

with simple exponential smoothing (SES) (e.g., Brown, 1963; Gardner, 2006), as this method is often used

in industry and also by the OEM which motivates our research. We ﬁrst discuss the design of the simu-

lation experiment (Section 3.1), and then we address the performance evaluation of the forecast methods

throughout the product life cycle (Section 3.2).

3.1. Experimental Design

We make the following assumptions for every period t(where tequals one week) in our simulation

experiment. First, machines containing the part under study are sold to customers. They become operational

immediately after the sales have occurred. We assume new sales occur following a Poisson process with rate

λ, although it is possible to use any kind of sales process. We consider two diﬀerent machine installation

rates λ={0.25; 1.25}installations per week. To mimic the product life cycle, we distinguish three stages in

the sales process. We let the expected sales be λ/2 in the ﬁrst 240 weeks, λin the next 400, and again λ/2

in the ﬁnal 160 weeks (until the sales end).

Second, when machines terminate their life, they no longer generate a spare part demand. We assume a

random machine survival probability and use an exponential distribution with parameter 1/ρto model the

time to discard, but any type of distribution can be included (exponentially distributed machine survival has

been used by for example Gharahasanlou et al., 2016; Hong et al., 2008; Hu et al., 2015; Kim et al., 2017;

Liu & Tang, 2016; Stormi et al., 2018). The average lifetime of a machine in our simulation experiment is

set to ρ= 720 weeks.

Third, maintenance is performed and parts are replaced. The time to failure probability distribution can

take any shape, although in a realistic setting it is positive and typically skewed to the right. Examples

are the lognormal distribution, the exponential distribution, or the Weibull distribution. We opt to use the

latter in this study, with scale and shape parameters αand β, as suggested by Barlow & Hunter (1960).

We consider two diﬀerent part failure properties α={336; 480}and β=1.5, assuming a machine operates

continuously (24/7). The scale and shape parameter of the part failure behaviour, αand β, are not aﬀected

by the product life cycle, as they are independent of the life cycle of the installed base. The preventive

maintenance interval is considered exogenous, and is set to τ= 2/3α.

Fourth, the installed base record is updated by monitoring and counting the parts in active machines

with its respective ages iand j. And ﬁfth, a demand forecast for the upcoming lead time is made. The lead

time Lvaries from 1 to 20 weeks. We set the smoothing constants for the SES and SBA forecast methods

equal to γ= 0.1, which is common practice for intermittent demand. As such, we investigate 80 scenarios

12

Parameter Values

λ{0.25; 1.25}per week

α{336; 480}

β1.5

ρ720 weeks

τ2/3 α

L{1,2,...,20}weeks

γ0.1

Table 2: Parameter settings of the numerical experiment.

Figure 2: Illustration of the link between the PLC of a machine, the installed base evolution, and the spare part demand per

period, together with the timings of the simulation experiment (Adapted from Dekker et al., 2013).

with parameter values inspired by the OEM and summarised in Table 2.

3.2. Performance Evaluation for Diﬀerent PLC Phases

We evaluate the model performance by registering the forecast accuracy, the achieved cycle service level,

and the average per period inventory holdings, in the three life cycle phases (initial, mature, and end-of-life).

Although the forecast method is identical in each of the three, we distinguish the performance evaluation in

these life cycle phases because the dynamics of spare part demand are related to those of the demand for

machines (Dekker et al. 2013). Figure 2 illustrates the connection between the product life cycle and spare

part demand. For the initial phase, we assume no previous knowledge on the part failure behaviour exists.

We can thus only start to make an estimate about the parameters of the part failure probability in this phase

as soon as the ﬁrst demand occurs. From then on, the model is periodically updated, i.e. we re-estimate

the failure behaviour based on new observations. We evaluate the model performance in the initial phase

during the ﬁrst 240 weeks, starting with the ﬁrst machine installation in week 1. The SBA method is also

initialised in this initial phase as proposed by Teunter & Duncan (2009). In the mature phase, we observe

the model performance from week 400 to 640, and in the EOL phase we observe week 1360 until week 1600.

13

Initial phase Mature phase EOL phase

λ α (1)

ADS

(2)

CV

(3)

APZ

(1)

ADS

(2)

CV

(3)

APZ

(1)

ADS

(2)

CV

(3)

APZ

0.25 336 1.02 0.04 97.85 1.1 0.27 83.15 1.08 0.25 87.06

480 1.01 0.01 98.69 1.07 0.22 88.69 1.05 0.18 90.66

1.25 336 1.1 0.26 89.1 1.55 0.51 39.77 1.4 0.47 49.88

480 1.05 0.15 93.61 1.32 0.45 56.32 1.27 0.42 61.92

Table 3: Descriptive demand statistics for β= 1.5 and ρ= 720 weeks: (1) The average per period demand size, when demand

occurs (ADS), (2) the coeﬃcient of variation of the per period demand size (CV), and (3) The average percentage of periods

with zero demand (APZ).

Figure 3: Example of a spare part demand time series, for λ= 1.25, α= 480, β= 1.5 and ρ= 720.

The performance of the methods is measured for every scenario in 100 simulation runs. Table 3 reports

some descriptive demand statistics for the diﬀerent scenarios, which shows the variety in demand patterns

in the diﬀerent scenarios: (1) The average demand size, when demand occurs (ADS), (2) the coeﬃcient

of variation of the demand size, when it occurs (CV ), and (3) the average percentage of weeks with zero

demand (AP Z). Figure 3 provides an example of a demand pattern for λ= 1.25, α= 480, β= 1.5, ρ= 720,

and the EOL phase starting after 800 weeks.

4. Results and Discussion

We ﬁrst discuss the performance of our proposed method when only corrective maintenance is in place.

We focus both on forecast accuracy (Section 4.1) as well as inventory implications (Section 4.2). Thereafter,

14

Initial phase Mature phase EOL phase

λ α SES SBA SMI SES SBA SMI SES SBA SMI

0.25 336 0.002 0.009 -0.004 0.005 0.061 -0.005 -0.001 -0.012 0

480 0.002 0.002 -0.005 0.004 0.059 0.003 -0.001 -0.012 0.001

1.25 336 0.011 0.062 -0.012 0.025 0.075 0.005 -0.009 0 0.004

480 0.007 0.037 -0.009 0.016 0.059 -0.003 -0.007 -0.008 0.003

Table 4: Mean Error (ME) measured in the diﬀerent PLC phases. The lowest absolute value for each scenario and PLC phase

is shaded. Additional parameter values: β= 1.5 and ρ= 720 weeks.

Initial phase Mature phase EOL phase

λ α SES SBA SMI SES SBA SMI SES SBA SMI

0.25 336 0.162 0.159 0.163 0.446 0.44 0.435 0.392 0.383 0.381

480 0.128 0.126 0.132 0.364 0.36 0.356 0.326 0.319 0.317

1.25 336 0.361 0.36 0.358 0.99 0.981 0.966 0.86 0.848 0.836

480 0.271 0.269 0.272 0.781 0.771 0.762 0.72 0.709 0.701

Table 5: Root Mean Square Error (RMSE) measured in the diﬀerent PLC phases. The lowest value for each scenario and PLC

phase is shaded. Additional parameter values: β= 1.5 and ρ= 720 weeks.

we discuss the performance of our method when a preventive maintenance policy is in use (Section 4.3).

4.1. Forecast Accuracy

We use the classic measures to evaluate forecast accuracy: Mean error (ME), and Root Mean Square

Error (RMSE). We do not include relative-to-the-series measures, such as the mean average percentage error,

because the actual demand (and thus the denominator in these forecast accuracy measures) is oftentimes

zero in intermittent demand patterns (Syntetos & Boylan, 2005). The results for the one-period-ahead ME

and RMSE are presented in Table 4 - 5. Our method using Service Maintenance Information is denoted as

SMI and is benchmarked against the Syntetos-Boylan approximation (SBA) and simple exponential smooth-

ing (SES).

We ﬁnd that the SMI method outperforms SES and SBA in the mature and EOL phase according to

both forecast accuracy measures, although the diﬀerence is limited. The SMI method is outperformed in the

initial phase of the PLC by SBA, but the diﬀerence is again very small.

We also observe that the SMI method tends to overestimate demand slightly in the initial phase, resulting

in a small negative ME. Given the large amount of zero values in the initial phase (between 89.1% and 98.69%

of the periods have zero demand), SES and SBA have a higher forecast accuracy in this phase.

Figure 4 shows the forecasting performance of the SMI and SBA methods for one particular scenario in the

mature phase. The SMI method, indicated by the dotted line, anticipates the increase in demand, whereas

the SBA method (the dashed line) lags behind. Moreover, the SBA method is very reactive whenever

a demand occurs, whereas the SMI methods provides a more stable forecast. Although not displayed in

Figure 4, similar behaviour can be observed in the EOL phase, where the SBA method tends to lag behind

15

Figure 4: The forecasting performance of SBA and SMI compared to the actual demand, for λ= 1.25, α= 480, β= 1.5 and

ρ= 720 in the mature phase.

on the decreasing trend in demand and consequently overestimates demand. This is also indicated by the

negative ME for SBA (and SES) in this phase.

4.2. Inventory Implications

In addition to the forecast performance, we also measure the impact of our SMI forecast method on the

inventory holdings. We use a periodic order-up-to-level (T, S ) policy, which is often used in an intermittent

demand context (e.g., Eaves & Kingsman, 2004; Sani & Kingsman, 1997; Syntetos & Boylan, 2006; Syntetos

et al., 2009b). At the end of the review period T, the demand forecast is estimated and the optimal order-

up-to-level Stis determined to satisfy a target service level. A replenishment order is placed to raise the

inventory position up to the level St.

We consider the inventory review period Tto be one week. During a week the following sequence of events

occurs: receive orders (placed L+1 weeks ago) at the beginning of the week, observe demand throughout

the week, forecast future demand (at the end of the week), evaluate the inventory position, and place a new

order if necessary.

The order-up-to level Stis optimised to meet a target cycle service level (CSL), which is the target

fraction of cycles in which demand can be delivered from stock. In our experiment, we attempt to meet a

CSL of respectively 70%, 80%, 85%; 90%, 95%, and 99%. Backorders can be carried forward. We use Eq. (8)

to deﬁne the forecasted lead time demand distribution, b

P(Dt,L =x), for the SMI method under corrective

16

Figure 5: Example of a cumulative lead time demand distribution at t= 600, when a corrective maintenance policy is in place,

with λ= 0.25, α= 336, β= 1.5, ρ= 720, L= 6 weeks and T= 1 .

maintenance. As such, Stis the smallest integer value that satisﬁes

CS L ≤

St

X

x=0 b

P(Dt,L =x).(12)

Figure 5 illustrates the forecasted lead time demand distribution for the scenario with λ= 0.25, α= 336,

L= 6, T= 1, at time t= 600, where a corrective maintenance policy is in place. For a target service level

of 70% and 99%, the required base-stock level equals 2 and 6 respectively.

To determine the order-up-to-level when the SES and SBA forecast methods are used, it is necessary to

make an assumption on the underlying demand distribution (Syntetos et al., 2009a). We follow the approach

of Babai et al. (2012), Syntetos & Boylan (2006), and Syntetos et al. (2015) to approximate the lead time

demand by a Negative Binomial Distribution (NBD). The NBD is deﬁned by a mean and a variance, which

can be deﬁned based on the forecasts. The mean of the distribution is estimated by the lead time (+ review

period) demand forecast. The variance is constructed by the associated smoothed mean squared error (MSE),

MSEt,T +L, with a smoothing constant ηof 0.25 (Babai et al., 2012; Pennings & Dalen, 2017; Syntetos et al.,

2010):

MSEt,T +L=η(b

Dt−(T+L),T +L−

t

X

i=t−T−L+1)2

+ (1 −η)MSEt−1,T +L,(13)

where Diis the demand occurring in period i. The NBD assumes the variance to be larger than the mean.

17

Figure 6: Tradeoﬀ between the achieved CSL and the average inventory levels, averaged over all scenarios, when a corrective

maintenance policy is in place.

When this assumption is not met, Sani & Kingsman (1997) show that the variance can be set equal to 1.1

times the mean. The order-up-to level for a target CSL is calculated as the corresponding quantile of the

lead time (plus review period) demand distribution.

We investigate the tradeoﬀ between the achieved CSL and the average per period inventory levels nec-

essary to attain this CSL. Figure 6 presents this tradeoﬀ curve over all scenarios, life cycle phases, and lead

times, and shows how the SMI method outperforms SES and SBA. As the tradeoﬀ curve associated with the

SMI method lies consistently above the frontier of SES and SBA, using the SMI forecast method results in

lower average inventory holdings than SES and SBA, for the same service level. Likewise, it yields a higher

service level for the same inventory holdings.

If we look into more detail, we can observe diﬀerences in performance between the PLC phases (Figure 7).

In the initial phase (Figure 7a), the SMI method is outperformed for most CSL targets by SBA and SES.

As indicated by the ME, SMI overestimates demand in the initial phase, and consequently reaches high

CSLs with more inventory than necessary. SES and SBA, on the contrary, underestimate the demand in

the initial phase according to the ME, resulting in lower inventory levels. Because of the highly sporadic

nature of demand in the initial phase (89 to 98% of periods without demand) SES nonetheless reaches the

target CSLs of 70%, and 80%. For a target CSL over 95%, however, the SMI method outperforms SES and

SBA in the initial phase, achieving a higher CSL with lower inventory levels. In the mature and EOL phase

18

(a) Initial phase (b) Mature phase

(c) EOL phase

Figure 7: Tradeoﬀ between the achieved CSL and the average inventory levels in the diﬀerent PLC phases, averaged over all

scenarios, when a corrective maintenance policy is in place.

(Figures 7b-7c), the SMI method outperforms SES and SBA consistently. In the EOL phase (Figure 7c) we

observe that for a target CSL of 99%, the SMI average inventory levels are 41% lower compared to SBA,

and 56% lower relative to SES.

In both the mature and EOL phase, we observe that the eﬃciency gains of the SMI method are higher for

longer lead times and a large installed base. Figure 8 illustrates the eﬃciency curves for the EOL phase for

short (L=1) and long (L=20) lead times, and small (λ=0.25) and large (λ=1.25) installed base sizes. The

gap between the SMI method and the SBA and SES method increases when Lincreases. The SMI method

anticipates the demand, whereas SBA and SES lag behind. This lagging eﬀect increases when lead times

are longer. Figure 8 also shows how the outperformance of the SMI method augments when the size of the

installed base increases. In this case, SMI has more information available to generate its forecasts.

19

(a) λ=0.25, L=1 (b) λ=0.25, L=20

(c) λ=1.25, L=1 (d) λ=1.25, L=20

Figure 8: Tradeoﬀ between the achieved CSL and the average inventory levels in the EOL phase, for diﬀerent values of lead

time and installed base size, with α=366, β=1.5, and ρ=720.

Similar behaviour can be observed in the mature phase, as illustrated in Figure 9. Here, we present the

results for three diﬀerent lead time scenarios (L=1, L=8, and L=20), when λ=1.25 and α=480. The SMI

method outperforms SBA and SES in terms of eﬃciency, and the diﬀerence in performance increases with a

longer lead time.

4.3. Preventive Maintenance

When a periodic preventive replacement policy is in place the SES and SBA methods can be applied in

two ways: Either we can use them to forecast all demand (both for corrective and preventive replacements),

as is often done in practice. Or the methods can be applied to only forecast the stochastic part of the demand,

i.e., the demand for corrective replacements, and separate it from the demand for preventive replacements,

20

(a) L=1 (b) L=8

(c) L=20

Figure 9: Tradeoﬀ between the achieved CSL and the average inventory levels in the mature phase, for diﬀerent lead time

values, with λ=1.25, α=480, β=1.5, and ρ=720.

21

Figure 10: Tradeoﬀ between the achieved CSL and the average inventory levels, averaged over all scenarios, when a preventive

maintenance policy is in place.

which is known some time in advance (see also Poppe et al., 2017). We refer to the former methods as SES

and SBA, and to the latter methods as SES PM and SBA PM.

Figure 10 shows the inventory-service tradeoﬀ curve of SES, SBA , SES PM and SBA PM compared with

our SMI method, averaged over all scenarios, life cycle phases, and lead times. It illustrates how SES PM

and SBA PM (continuous lines) outperform their counterparts which do not include the advance demand

information (dashed lines). The SMI method outperforms all methods.

Looking at the diﬀerent lifecycle phases, we ﬁnd that in the initial phase (Figure 11a) the SMI method

only outperforms the other forecast methods for high target CSLs. This was also observed when a corrective

maintenance policy was in place (Figure 7a). In the mature and EOL phase (Figure 11b and 11c) the SMI

method outperforms SBA and SES, even when advance demand information of the preventive replacements

is taken into account in their forecasts (SES PM and SBA PM).

5. Conclusions

In this paper we present a method to forecast spare part demand which incorporates information on the

service maintenance operations that drive spare part demand. The main idea behind our approach is that

spare part demand is generated by part failures and equipment maintenance, which is useful information to

predict future demand. We consider a larger set of spare part demand drivers (in combination with each

other) than the existing literature to construct the forecasts: information on the (evolution of the) active

installed base, the part reliability, and the maintenance policy to forecast future spare part demand.

We provide a forecast of the distribution of the future spare parts demand during the upcoming lead

22

(a) Initial phase (b) Mature phase

(c) EOL phase

Figure 11: Tradeoﬀ between the achieved CSL and the average monthly inventory levels in the diﬀerent PLC phases, when a

preventive maintenance policy is in place.

23

time, rather than a point forecast. This distribution is in turn used to manage inventories; We apply it to

determine the optimal a base-stock policy to attain a pre-deﬁned target service level. This way, we connect

forecasting directly with inventory management.

Our approach is intuitively appealing and straightforward, which can facilitate its implementation in

practice. Additionally, our method is applicable throughout the full product life cycle of the installed

machines, which means there is no need to use diﬀerent methods for each of the PLC phases, and there is

also no need to identify each PLC stage. Moreover, it can be applied both when both a corrective and a

preventive maintenance policy apply, and it is not dependent on predeﬁned probability distributions.

A numerical simulation experiment shows its improvement potential compared to the current best-in-

class forecast methods that do not take this service maintenance information into account. Our method is

capable to achieve higher cycle service levels with lower inventories in the mature and end-of-life phase of

the machines, and its improvement potential is larger when the installed base is relatively large, and when

lead times are longer.

The main challenge to implement our method is the collection of accurate data. Our method requires

keeping track of historical machine sales and discards to monitor the evolution of the installed base over

time, as well as a history of past part failures, and information on (past and future) preventive maintenance

interventions. As such, forecasting becomes the outcome of an inter-organizational process, where cooper-

ation and information sharing (for example on historical failures, maintenance actions, and product sales)

between diﬀerent departments within the same company is needed. When the quality of the data is poor,

the method lacks its necessary inputs.

A second limitation of our method is that it does not consider the possibility of a part requiring mul-

tiple replacements within the same replenishment lead time, where the replaced part (either preventive or

corrective) fails again prior to its replenishment. This renders the model less accurate with very long lead

times (multiple months) in combination with short part lifetimes (failure within the month). In most set-

tings, including ours, the time to failure is typically much longer than the replenishment lead time. Heavy

equipment typically has an expected lifetime of several years, where the probability of the two replacements

within the same replenishment lead time is negligible.

For future work we suggest to investigate the impact of imperfect or missing information on the re-

sults, given the sensitivity of our method to good data quality. We could also extend the model with

inspection-based preventive maintenance, where the demand originating from preventive inspections is con-

sidered stochastic. Finally, whereas this paper optimises inventories for a given target service level, future

research could be devoted to a dynamic inventory model with cost-optimal service levels.

24

References

Auramo, J., & Ala-risku, T. (2005). Challenges for going downstream. International Journal of Logistics:

Research and Applications,8, 333–345.

Babai, M. Z., Ali, M. M., & Nikolopoulos, K. (2012). Impact of temporal aggregation on stock control

performance of intermittent demand estimators : Empirical analysis. Omega,40 , 713–721. doi:10.1016/

j.omega.2011.09.004.

Bacchetti, A., & Saccani, N. (2012). Spare parts classiﬁcation and demand forecasting for stock control:

Investigating the gap between research and practice. Omega,40 , 722–737.

Barabadi, A. (2012). Reliability and spare parts provision considering operational environment: A case

study. International Journal of Performability Engineering,8, 497–506.

Barlow, R., & Hunter, L. (1960). Optimum Preventive Maintenance Policies. Operations Research,8,

90–100.

Boylan, J. E., & Syntetos, A. A. (2010). Spare parts management: A review of forecasting research and

extensions. IMA Journal of Management Mathematics,21 , 227–237.

Brown, R. G. (1963). Smoothing, Forecasting and Prediction of Discrete Time Series. Englewood Cliﬀs,

New Jersey: Prenctice-Hall.

Cohen, M. A., Agrawal, N., & Agrawal, V. (2006). Winning in the aftermarket. Harvard Business Review,

(pp. 129–138).

Croston, J. D. (1972). Forecasting and Stock Control for Intermittent Demands. Operational Research

Quarterly,23 , 289–303.

Dekker, R., Pin¸ce, C¸ ., Zuidwijk, R., & Jalil, M. N. (2013). On the use of installed base information for spare

parts logistics: A review of ideas and industry practice. International Journal of Production Economics,

143 , 536–545.

Eaves, A. H. C., & Kingsman, B. G. (2004). Forecasting for the Ordering and Stock-Holding of Spare Parts.

The Journal of the Operational Research Society,55 , 431–437. doi:10.1057/palgravejors.2601390.

Gardner, E. S. (2006). Exponential smoothing : The state of the art — Part II. International Journal of

Forecasting,22 , 637–666. doi:10.1016/j.ijforecast.2006.03.005.

25

Gharahasanlou, A. N., Ataei, M., Khalokakaie, R., Ghodrati, B., & Jafarie, R. (2016). Tire demand planning

based on reliability and operating environment. International Journal of Mining and Geo-Engineering,

50 , 239–248.

Hasni, M., Aguir, M. S., Babai, M. Z., & Jemai, Z. (2018). Spare parts demand forecasting : a review on

bootstrapping methods. International Journal of Production Research, . doi:10.1080/00207543.2018.

1424375.

Hong, J. S., Koo, H.-y., Lee, C.-s., & Ahn, J. (2008). Forecasting service parts demand for a discontinued

product. IIE Transactions,40 , 640–649.

Hong, Y. (2013). On computing the distribution function for the Poisson binomial distribution. Computa-

tional Statistics and Data Analysis,59 , 41–51.

Hu, Q., Bai, Y., Zhao, J., & Cao, W. (2015). Modeling spare parts demands forecast under two-dimensional

preventive maintenance policy. Mathematical Problems in Engineering,2015 .

Jalil, M. N., Zuidwijk, R., Fleischmann, M., & van Nunen, J. A. (2011). Spare parts logistics and installed

base information. Journal of the Operational Research Society,62 , 442–457.

Jardine, A. K. S., & Tsang, A. H. (2013). Maintenance, replacement, and reliability: theory and applications.

CRC press.

Jin, T., & Liao, H. (2009). Spare parts inventory control considering stochastic growth of an installed base.

Computers and Industrial Engineering,56 , 452–460.

Kim, T. Y., Dekker, R., & Heij, C. (2017). Spare part demand forecasting for consumer goods using installed

base information. Computers and Industrial Engineering,103 , 201–215.

Liu, X., & Tang, L. C. (2016). Reliability analysis and spares provisioning for repairable systems with

dependent failure processes and a time-varying installed base. IIE Transactions,48 , 43–56.

Lu, X.-C., & Wang, H.-N. (2015). The Laptop Spare Parts Studying under Considering Users’ Repair

Willingness. International Journal of Simulation Modelling,14 , 158–169.

Minner, S. (2011). Forecasting and Inventory Management for spare Parts: An Installed Base Approach. In

N. Altay, & L. A. Litteral (Eds.), Service Parts Management Chapter 8. (pp. 157–169). London: Springer.

Pennings, C. L. P., & Dalen, J. V. (2017). Integrated hierarchical forecasting. European Journal of Opera-

tional Research,263 , 412–418.

26

Poppe, J., Basten, R. J. I., Boute, R. N., & Lambrecht, M. R. (2017). Numerical study of inventory

management under various maintenance policies. Reliability Engineering and System Safety,168 , 262–

273.

Ritchie, E., & Wilcox, P. (1977). Renewal theory forecastting for stock control. European Journal of

Operational Research,1, 90–93.

Romeijnders, W., Teunter, R., & Van Jaarsveld, W. (2012). A two-step method for forecasting spare parts

demand using information on component repairs. European Journal of Operational Research,220 , 386–

393.

Sani, B., & Kingsman, B. G. (1997). Selecting the Best Periodic Inventory Control and Demand Forecasting

Methods for Low Demand Items. Journal of the Operational Research Society,48 , 700–713. doi:10.1057/

palgrave.jors.2600418.

Si, X.-s., Zhang, Z.-X., & Hu, C.-H. (2017). An adaptive Spare Parts Demand Forecasting Method Based

on Degradation Modeling. In Data-Driven Remaining Useful Life Prognosis Techniques Chapter 15. (pp.

405–417). Berlin: Springer.

Stevenson, W. J. (2012). Operations Management: Theory and Practice. New York: McGraw-Hill/Irwin.

Stormi, K., Laine, T., Suomala, P., & Elomaa, T. (2018). Forecasting sales in industrial services: Modeling

business potential with installed base information. Journal of Service Management,29 , 277–300.

Syntetos, A. A., & Boylan, J. E. (2005). The accuracy of intermittent demand estimates. International

Journal of Forecasting,21 , 303–314.

Syntetos, A. A., & Boylan, J. E. (2006). On the stock control performance of intermittent demand estimators.

International Journal of Production Economics,103 , 36–47. doi:10.1016/j.ijpe.2005.04.004.

Syntetos, A. A., Boylan, J. E., & Disney, S. M. (2009a). Forecasting for inventory planning: a 50-year review.

Journal of the Operational Research Society,60 , 149–161.

Syntetos, A. A., Nikolopoulos, K., & Boylan, J. E. (2010). Judging the judges through accuracy-implication

metrics : The case of inventory forecasting. International Journal of Forecasting,26 , 134–143. doi:10.

1016/j.ijforecast.2009.05.016.

Syntetos, A. A., Nikolopoulos, K., Boylan, J. E., Fildes, R., & Goodwin, P. (2009b). The eﬀects of inte-

grating management judgement into intermittent demand forecasts. International Journal of Production

Economics,118 , 72–81.

27

Syntetos, A. A., Zied Babai, M., & Gardner, E. S. (2015). Forecasting intermittent inventory demands:

Simple parametric methods vs. bootstrapping. Journal of Business Research,68 , 1746–1752. doi:10.

1016/j.jbusres.2015.03.034.

Teunter, R. H., & Duncan, L. (2009). Forecasting intermittent demand: a comparative study. The Journal

of the Operational Research Society,60 , 321–329. doi:10.1057/palgrave.jors.2602569.

Van der Auweraer, S., Boute, R. N., & Syntetos, A. A. (2019). Forecasting spare part demand with in-

stalled base information : A review. International Journal of Forecasting ,35 , 181–196. doi:10.1016/j.

ijforecast.2018.09.002.

Wang, W. (2011). A joint spare part and maintenance inspection optimisation model using the Delay-Time

concept. Reliability Engineering and System Safety,96 , 1535–1541.

Wang, W., & Syntetos, A. A. (2011). Spare parts demand: Linking forecasting to equipment maintenance.

Transportation Research Part E ,47 , 1194–1209.

Wu, X., & Bian, W. (2015). Demand Analysis and Forecast for Spare Parts of Perishable Hi-Tech Products.

In Logistics, Informatics and Service Sciences (LISS), 2015 International Conference on (pp. 1–6). IEE.

28