Forecasting spare part demand using service maintenance information

Abstract

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

Full text

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
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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).
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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,
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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
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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.
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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.
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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

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