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We describe a data-driven approach to optimize the periodic maintenance policies for a portfolio with different machine profiles. As insufficient data may be available per profile to make an accurate assessment of the failure intensities and costs, we make use of data pooling to alleviate the small data problem. This may reduce maintenance costs compared to a stratified approach that splits the data in subsets per machine profile, and compared to a uniform approach that treats all profiles the same.
Data-driven preventive maintenance for a heterogeneous machine portfolio
Laurens Deprez1, Katrien Antonio2,3, Joachim Arts1, and Robert Boute2,4,5
1Luxembourg Centre for Logistics and Supply Chain Management, University of Luxembourg, Luxembourg.
2Faculty of Economics and Business, KU Leuven, Belgium.
3Faculty of Economics and Business, University of Amsterdam, The Netherlands.
4Technology & Operations Management Area, Vlerick Business School, Belgium.
5VCCM, Flanders Make, Belgium.
January 6, 2023
We describe a data-driven approach to optimize periodic maintenance policies for a heterogeneous port-
folio with different machine profiles. When insufficient data are available per profile to assess failure
intensities and costs accurately, we pool the data of all machine profiles and evaluate the effect of (ob-
servable) machine characteristics by calibrating appropriate statistical models. This reduces maintenance
costs compared to a stratified approach that splits the data into subsets per profile and a uniform ap-
proach that treats all profiles the same.
Keywords: preventive maintenance, data pooling, proportional hazards, small data
1 Introduction
Despite advances in condition monitoring, time- or age-based periodic preventive maintenance is still com-
mon practice in many companies. Under this policy, preventive maintenance (PM) interventions are sched-
uled at a periodic interval, either measured in calendar time or running hours. This interval is determined by
trading off the failure rate and corresponding failure costs against the cost of performing preventive mainte-
nance. Most literature assumes that the failure behaviour, or failure intensity, is known, in line with Barlow
and Hunter [1960]. In practice, however, the failure distribution may not be known and should be estimated
based on historical data. This may introduce some challenges. A small machine population may not provide
sufficient maintenance and failure data for an accurate estimation. The lack of sufficient data to accurately
estimate regression parameters is known as the ‘small data’ problem. Even when the machine population is
of adequate size, the machine population might be heterogeneous and as such may contain multiple different
machine profiles. This heterogeneity can be caused by different running conditions, environmental factors
or different manufacturing plants where the machines were assembled and might induce different failure
intensities and costs. Estimation by stratifying, or splitting the data, per machine profile might again induce
the small data problem. Aggregating these machine data without taking into account the heterogeneity in
the machine population leads to a maintenance policy that is optimal for an average machine, but it is not
tailored to a particular machine (profile). The novelty of our approach is that we introduce a multivariate
time-to-failure and cost model such that the data of all machines (across the different machine profiles) can
be pooled. This allows differentiating the PM policy over the different machine profiles, while at same time
To appear in Operations Research Letters
corresponding author:, 6 Rue Richard Coudenhove-Kalergi, 1359 Luxembourg, Luxembourg
obtains accurate time-to-failure and maintenance cost estimates for a heterogeneous machine population.
The latter is not possible when stratifying data per machine profile.
Service providers, who maintain the assets of their customers, can make use of such an approach. Their
maintenance portfolio provides them potentially with ample historical maintenance and failure data, yet
data per machine profile might be limited. Our objective is to optimize the periodic maintenance interval
by explicitly considering the heterogeneity in the machine portfolio that is induced by observable machine
characteristics. Our policy optimization applies to a finite horizon, motivated by the finite duration of service
maintenance contracts or (extended) warranties [Nakagawa and Mizutani,2009;Dursun et al.,2022]. We
focus on observable machine characteristics that remain constant during the contract horizon, e.g. running
conditions, country, operating environment and industry type. This categorizes each machine in a specific
machine profile, specified by these machine characteristics. By tailoring the maintenance policy to the
machine profile, the resulting cost optimization may lead to higher profit margins or lower contract prices.
Population heterogeneity has been studied by Dursun et al. [2022], Abdul-Malak et al. [2019], and
de Jonge et al. [2015], among others. In Dursun et al. [2022]; Abdul-Malak et al. [2019], and de Jonge et al.
[2015], parts (or machines) originate from multiple, mostly two, different sub-populations with different fail-
ure distributions. It is unknown to which sub-population a spare part belongs but the failure distribution of
each sub-population itself is known. In their analysis, the sub-population of the part is inferred by observing
its failure behaviour and adapting the maintenance policy accordingly. In contrast, we consider the case
where it is known to which sub-population machines belong as machines are labeled by their machine pro-
file, characterized by observable machine characteristics. The underlying failure distribution, however, is not
known and we infer the failure behaviour and maintenance costs for each machine profile from data. Drent
et al. [2020] also deals with an unknown failure distribution but assumes population homogeneity. In their
paper the failure distribution and maintenance policy are inferred by means of a Bayesian approach as infor-
mation accumulates during the machine operation. The information accumulation process consists of both
censored (i.e., preventive replacements) and uncensored (i.e., corrective replacements) observations of the un-
derlying lifetime distribution. This leads to an inherent exploration-exploitation trade-off. Exploration (i.e.,
a longer periodic maintenance interval) increases the probability of corrective replacement, but at the same
time leads to accumulating more, valuable, information. In our approach there is no exploration-exploitation
trade-off as the data over all machine profiles has been collected prior to the moment of estimation.
Our methodology relies on a data set with historical failure and maintenance data of a heterogeneous
machine portfolio. We learn the failure behaviour and costs and asses the effect of (observable) machine
characteristics by calibrating appropriate statistical models. The calibrated statistical models enable the
optimization of each machine profile’s periodic maintenance interval. We study how the resulting mainte-
nance policies are more cost-effective than (1) a uniform approach that disregards the machine profiles, and
(2) a stratified approach that splits the data per machine profile to take into account each machine profile.
The next section introduces our reliability model, the optimality condition for the periodic maintenance
interval and the predictive models to calibrate the failure behaviour and costs. Section 3numerically eval-
uates how and when our data pooling approach reduces maintenance costs. Section 4concludes.
2 Maintenance policies for a heterogeneous machine population
We first set out the details of our reliability model. We then specify the optimality condition for a periodic,
preventive maintenance policy over a finite horizon for a heterogeneous machine population. Finally we
establish the statistical models and their calibration to data to estimate the failure rate, and the costs of
failure and preventive maintenance.
2.1 Reliability model
We consider the optimization of the number of preventive maintenance interventions, n, for a machine
during a finite horizon [0,t], e.g. the coverage period of a service agreement. In our analysis we will
assume that failures require minimal, as-good-as-old, corrective maintenance [see e.g. Barlow and Proschan,
1965;Lindqvist et al.,2006;Wu and Zuo,2010;Doyen and Gaudoin,2011;Arts and Basten,2018] and
planned, preventive maintenance actions are perfect. The latter justifies a periodic maintenance policy, where
preventive maintenance is executed with a fixed interval tpm =t
n+1 , at times tk=ktpm (k= 1, ..., n).
Each machine is characterized by a set of machine characteristics or covariates x, e.g. operating conditions,
industry type, or country, that remain the same during the planning horizon.
Failure times We specify the machine-dependent failure intensity function λ(t) in each maintenance in-
terval under the Cox proportional hazards assumption [Cox,1975] with the same baseline failure intensity
function λ0(t) for each machine and βrepresenting the impact of the machine characteristics x. The pro-
portional hazards model is a versatile model to analyze the effect of operating conditions (or covariates) on
the lifetime of a system. Kumar and Westberg [1997] have shown that models from the proportional hazards
family appear to be the better ones for analyzing the effect of the covariates. It is also very convenient to
add terms or cross products. The practical value of the proportional hazards model has been demonstrated,
for instance, by Barabadi et al. [2014] in a case study in mining equipment to identify the covariates that in-
fluence the reliability. We acknowledge that, like any multi-variate model, there is a risk of mis-specification
in proportional hazard models. We show in Section 3.5 that the benefits of data pooling outweigh this risk.
The machine-dependent failure intensity function λ(t) defines a non-homogeneous Poisson process [Moller
and Waagepetersen,2003] for the arrival of failures. With tthe time since the last PM intervention, the
machine-specific failure intensity function λ(t) is then characterized by
λ(t) = λ0(t) exp(β·x) for t[0, tktk1) (k),(1)
where t0= 0 and tn+1 = t, respectively referring to the start and end of the planning horizon. Assuming
perfect preventive maintenance, the failure intensity function λ(t) is the same for each inter-PM interval.
The baseline failure intensity λ0(t) can take any parametric form. We define λ0(t) by a Weibull failure
intensity function, which is regularly used in the reliability literature to model failure times [Bobbio et al.,
1980;Wang et al.,2000;Wu,2019], with scale parameter αR+
0and shape γR+,
λ0(t) = γαγtγ1.
Costs of maintenance The expected preventive maintenance cost, cp(x), and expected failure cost,
cf(x) are modeled with a gamma generalized linear model (GLM) to account for the impact of the machine
characteristics x. [Delong et al.,2021] argues that gamma models are appropriate to model cost data given
their positive support and right-skewed distribution. We denote the scale of the gamma distribution by θf
for the failure costs. For the expected failure costs, the GLM with categorical explanatory variables xis
specified by [see Ohlsson and Johansson,2010, for an overview]
cf(x) = exp β
f·(1,x)= exp
βf,j xj
with β
f·(1,x) the linear predictor. The impact of the machine characteristics xon the cost of failure is
captured by βf. We remark that the first component of the vector βfacts as an intercept and consequently
the length of the vector βfis one plus the length of vector x, i.e. 1+ q. The expected preventive maintenance
costs cp(x) are similarly specified. Their respective parameters are denoted with subscript p. We remark
that we set the parameters for the cost of preventive maintenance cp(x) and the cost of failure cf(x) such
that cp(x)< cf(x) for each combination of machine characteristics x.
Total maintenance costs The expected total maintenance cost over the horizon [0,t] for a policy with
preventive maintenance at (consecutive) times {t1, t2, ..., tn}([0,t]), on a machine with characteristics
xis then defined by,
Cx({t1, t2, ..., tn}) = cf(x)
k=0 Ztk+1tk
λ(t)dt +ncp(x)
k=0 Ztk+1tk
λ0(t) exp(β·x)dt +ncp(x)
=cf(x) exp(β·x)
Λ0(tk+1 tk) + ncp(x),
where Λ0(t) = Rt
0λ0(u)du. The integral Rtk+1tk
0λ(u)du represents the expected number of failures between
tkand tk+1. The latter is a property of non-homogeneous Poisson processes Lewis and Shedler [1979].
By summing over all preventive maintenance intervals we obtain all failures during the horizon [0,t].
Equation (3) can be simplified for periodic maintenance with fixed interval tpm =t
n+1 . In this case, the
expected total maintenance costs Cx(n) is a function of the number of preventive maintenance interventions,
Cx(n) = cf(x) exp(β·x)(n+ 1)Λ0(∆tpm) + ncp(x).(4)
Observe that the impact of the machine characteristics xon the failure rate, i.e. exp(β·x) is equivalent to
an increase in the expected failure costs with the same factor, exp(β·x).
2.2 Optimality condition for a differentiated, periodic policy
We denote nthe optimal number of PMs that minimizes the total maintenance costs in Eq. (4) during the
planning horizon [0,t]. Although ndepends on the machine characteristics x, we will adopt ninstead
of n(x) to simplify notation.
Proposition 1. If the baseline failure intensity λ0(t)is strictly increasing in t(R+), then the optimal
number of PMs, n, is the smallest nN0that satisfies,
(n+ 1)Λ0t
n+ 1(n+ 2)Λ0t
n+ 2cp(x)
cf(x) exp(β·x).(5)
Proof. The second order derivative of the total maintenance cost (for nR+) is
dn2=cf(x) exp(β·x)t2
(n+ 1)3λ
n+ 1,
with λ
0(t) the first order derivative of λ0(t). If λ0(t) is strictly increasing, i.e. λ
0(t)>0 (tR+), then Cx(n)
is convex on R+. The optimal nsatisfies the first order optimality condition, i.e. it is the smallest n(N0)
for which 1[Cx](n)0 where the forward difference of the cost function 1[Cx](n) = Cx(n+1)Cx(n).
2.3 Calibration of predictive models to estimate failure intensity and costs
The optimal periodic maintenance policy depends on the failure intensity and the costs of failure and
maintenance of each machine profile (as defined by the characteristics x). We now describe the statistical
models and their calibration to estimate this failure intensity and the costs given a data set with failure and
maintenance records.
Time-to-failure model From the failure intensity function for each inter-PM interval, λ(t), (Eq. (1)) and
the timings of the PM interventions, {t1, t2, ..., tn}, we define the failure intensity function λT(t) in absolute
time, i.e. since the start of the observation horizon,
λT(t) =
λ0(t) exp(β·x) if 0 t<t1
λ0(tt1) exp(β·x) if t1t<t2
λ0(tt2) exp(β·x) if t2t<t3
λ0(ttn) exp(β·x) if tntt.
The timings of the PM interventions, or the PM interval, should be known to set up the expression for
λT(t). This is not an issue, however, since the PM interventions are planned upfront. Denote R(t|t) the
reliability, or survival function, for the time-to-next-failure, with tthe time of the previous failure,
R(t|t) = exp Zt
To calibrate the time-to-failure model and consequently find estimates for the parameters of the baseline
failure intensity function λ0(t), i.e. αand γ, and the impact of the covariates β, we maximize the time-to-
failure log-likelihood. The events of interest are the failures as well as the end of the observation horizon for
each machine in the data. The latter acts as a censoring event. Each event is characterized by the vector
(tf, tf,,x, δ), where tfis the event time, tf,the time of the previous event, xthe machine characteristics
and δ {0,1}where δ= 0 indicates a censored event, i.e. the end of the observation horizon, and δ= 1 a
failure. Summing over failure events jon machine iin the data provides the expression for the log-likelihood
for the time-to-failure data,
L(α, γ, β) =
δjlog (λT(tf,j )) + log (R(tf,j |tf,j,)) ,(6)
where Nis the total number of machines and fiis the number of failures (including the end of the observation
horizon) on machine i. Consequently, the failures contribute with the logarithm of the probability density
function and the end of the observation horizon with the logarithm of the reliability. Maximizing the log-
likelihood L(α, γ, β) for the time-to-failure model leads to estimates for the parameters αand γof the
baseline failure intensity function λ0(t) and the impact of the covariates β.
Costs model We calibrate separate gamma generalized linear models (GLMs), as specified in Eq. (2), for
the expected preventive maintenance costs cp(x) and the expected failure costs cf(x) taking into account the
machine characteristics x. This provides estimates for βpand βf, the impact of the machine characteristics
xon the preventive maintenance costs and on the failure costs respectively, and for θpand θf, the scale
parameter of the gamma distribution of the preventive maintenance costs and of the failure costs respectively.
Benchmark approaches The approach described above makes use of all available failure and mainte-
nance data by pooling the data across all machines. We therefore refer to this approach as the pooling
approach. Yet, by specifying and calibrating the impact of the machine characteristics xon the failure in-
tensity and costs, our approach can differentiate the optimal periodic policy per machine profile. We specify
two benchmark approaches. First, a uniform approach that aggregates the data, but disregards the machine
profiles, calibrates the time-to-failure and costs models ignoring the machine characteristics x. This is iden-
tical to setting β= 0 when optimizing the log-likelihood in Eq. (6). Similarly, we also ignore the machine
characteristics xwhen calibrating the cost models. This gives us only estimates for βp,0, θp, βf,0and θp.
This approach also makes use of all available data, but the models are calibrated as if all machines would
have no machine-specific characteristics, resulting in a uniform PM policy with identical, optimized number
of PM interventions nfor each machine (profile). Second, we also benchmark against a stratified approach.
For this approach, we split or stratify the data in subsets that only contain data on a single machine profile,
i.e. combination of machine characteristics, and then calibrate the time-to-failure and costs models in similar
fashion to the uniform approach, i.e. by ignoring the machine characteristics x, for each subset. The models
for each machine profile serve as input to optimize the number of PM interventions. Although these policies
are capable of differentiating over the different machine profiles, they use less data to calibrate the models.
The latter can lead to less accurate estimates and inferior cost performance of the resulting maintenance
policies. In our numerical analysis we will also benchmark against the oracle approach. The oracle knows
the distribution of the failure behaviour and costs, and their associated parameters and is hence equivalent
to assuming an infinite number of observations. Here we use the true distributions (rather than estimates)
to find the optimal number of preventive maintenance interventions for each machine profile. These oracle
policies serve as a lower bound on the costs.
3 Results and insights
We set up a numerical experiment to assess the value of differentiating the PM policy per machine profile by
pooling the data using the approach described in the previous section. To do so, we generate a data set with
maintenance and failure records from a heterogeneous machine portfolio with different machine profiles. The
simulation engine to generate this data set is described in Section 3.1. We calibrate the parameters of the
time-to-failure and cost models making use of maximum likelihood estimation (described in Section 2.3), and
apply the optimality condition (Proposition 1) to prescribe the optimal number of preventive maintenance
interventions for each machine profile. We report the cost performance of this approach in Section 3.2 and
illustrate how each approach performs with limited amounts of data in Section 3.3. In Section 3.4 we check
whether it is actually worth differentiating the PM policy at all compared to adopting a uniform PM policy
that is identical across all machine profiles. Finally, Section 3.5 studies model mis-specification where the
fitted model has a different parametric form from the model from which the data were simulated.
3.1 Simulation engine
We generate a data set of failures and maintenance records for a heterogeneous machine portfolio, following
the reliability model introduced in Section 2.1. We consider a portfolio with 240 machines, of which 90%
is observed for ti= 5 years and 10% has a shorter history of ti
U[1,5] years. We characterize each
machine by 4 features, x= (x1,1, x1,2, x1,3, x1,4)T {0,1}4which are randomly assigned following a uniform
distribution. This leads to 16 different machine profiles, each occurring in the portfolio with equal probability.
The preventive maintenance interventions in the data set are executed periodically with an interval of
tPM = 1 year. Recurrent failure times in a PM interval are generated by inverse transform sampling from
the failure distribution determined by the failure intensity function in Eq. (1) [Metcalfe and Thompson,
2006;Cook and Lawless,2007;Jahn-Eimermacher et al.,2015;enichoux et al.,2015]. The costs of failures
and preventive maintenance interventions are positive, follow a right-skewed distribution and are dependent
of machine characteristics x. To accommodate for these properties, they are sampled from a gamma GLM
[Denuit et al.,2007;De Jong et al.,2008].
The simulation parameters that we have used to generated failure times and costs are summarized in
Table 3(Appendix A). Our parameters set the mean time-to-failure for the machine profile x= (0000) equal
to 1.27 years, if the machine would not be maintained preventively. The expected costs cp(x) and cf(x) for
machine profile x= (0000) are respectively 30 and 300. Without loss of generality, we let the costs of a PM
machine profile oracle pooling stratified uniform
0 0 0 0 10 9 11 10
0 0 0 1 6 6 5 10
0 0 1 0 8 8 7 10
0 0 1 1 5 5 4 10
0 1 0 0 13 11 6 10
0 1 0 1 9 8 6 10
0 1 1 0 11 9 9 10
0 1 1 1 7 6 5 10
1 0 0 0 14 13 13 10
1 0 0 1 9 9 10 10
1 0 1 0 11 11 13 10
1 0 1 1 7 7 5 10
1 1 0 0 18 15 21 10
1 1 0 1 12 11 10 10
1 1 1 0 15 13 15 10
1 1 1 1 10 9 5 10
Table 1: Prescribed number of PMs during a contract horizon of 5 years for a single data set.
be independent of the machine characteristics, as the optimality condition (Eq. (5)) only depends on the
ratio of cp(x) and cf(x). In Table 4(Appendix B), we display an extract of the failure and maintenance
records in our simulated data set. In Sections 3.2-3.4, the data are generated from correctly specified models,
i.e., of the same parametric form. Section 3.5 studies model mis-specification where the fitted model has a
different parametric form from the model from which the data were simulated.
3.2 What is the value of data pooling?
Table 1reports for a single simulated data set the optimal number of preventive maintenance interventions,
n, during a time horizon of t= 5 years for each of the 16 machine profiles, as determined by the
different approaches. These different approaches all rely on the optimality condition in Proposition 1, yet
with different estimations of the failure behaviour and the costs. We refer to Section 2.3 for a discussion
of these approaches. Depending on the (simulated) data set, the estimations of the failure behaviour and
the costs, and therefore also the prescribed number of PM interventions, may slightly differ. To report the
cost performance of each of these approaches, we therefore generated 100 data sets with identical simulation
parameters. Table 2reports the average total maintenance costs for each approach (making use of Eq. (4)),
as well as the empirical 95%-confidence interval over the 100 simulation runs. The costs are given with
respect to the costs of the oracle approach by means of a multiplicative ratio. Furthermore, we include the
cost performance for an average machine profile. The maintenance cost for the average machine profile is
determined by taking the average of the costs of all machine profiles with equal weights. Since the oracle
serves as a lower bound, all other approaches will have higher or at best equal costs. The oracle provides
the optimal differentiated maintenance strategy over the different machine profiles. To obtain these policies,
however, we would need infinite amount of data to exactly know the underlying failure behaviour and costs.
The maintenance costs obtained under the stratified approach are on average only 5% higher compared
to the oracle for the average machine profile (see Table 2). However, for machine profile x= (1100) the
cost performance resulting from a stratified approach is worse, i.e. 8.3% on average, compared to the oracle.
For a specific data set, we also observe that there can be a large discrepancy between the prescribed number
of PMs by the oracle and the stratified approach, e.g. the oracle and the stratified approach respectively
prescribe 13 and 6 PMs for machine profile x= (0100) (see Table 1). In general, there is also a lot of spread
in the cost performance under the stratified approach. The 97.5% quantiles for the maintenance costs are
very high, up to 74.4% higher than the costs obtained by the oracle (see profile x= (0011) in Table 2).
machine profile oracle pooling (%) stratified (%) uniform (%)
0 0 0 0 634.09 100.5 (100,102.2) 104.2 (100,120.3) 100.3 (100,102.2)
0 0 0 1 415.9 100.2 (100,102.2) 105 (100,134.5) 109.5 (101.8,112.4)
0 0 1 0 513.71 100.3 (100,100.8) 104.6 (100,137.4) 102.8 (100,104.2)
0 0 1 1 334.48 100.3 (100,102.1) 107.5 (100,174.4) 121.6 (108.5,126.2)
0 1 0 0 822.79 100.9 (100,105) 105 (100,134.4) 103.2 (101.5,111)
0 1 0 1 542.25 100.3 (100,101.5) 105.6 (100,132.8) 101.6 (100,102.7)
0 1 1 0 668.46 100.6 (100,103.4) 103.7 (100,123.7) 100.3 (100,103.4)
0 1 1 1 438.12 100.3 (100,100.6) 104.9 (100,139.3) 107.5 (101.1,110)
1 0 0 0 866.42 100.9 (100,104.9) 103.7 (100,126.3) 104.6 (102.5,113.6)
1 0 0 1 570.88 100.3 (100,100.6) 103.8 (100,114.1) 101 (100,101.7)
1 0 1 0 704.05 100.6 (100,102.1) 103.9 (100,127.6) 100.7 (100,104.9)
1 0 1 1 462.11 100.3 (100,101.3) 104.2 (100,128.6) 105.6 (100.4,107.8)
1 1 0 0 1121.07 101.8 (100,111.5) 108.3 (100,136.6) 115.4 (111.5,130.8)
1 1 0 1 741.59 100.6 (100,103.3) 103.5 (100,119.8) 101.3 (100.2,106.7)
1 1 1 0 912.53 101.2 (100,106.6) 106.5 (100,135.6) 106.2 (103.7,116.4)
1 1 1 1 602.3 100.3 (100,101.2) 103.3 (100,116.9) 100.4 (100,101.2)
average 646.92 100.7 (100,103.5) 105 (101.1,111.7) 105 (104.6,108.5)
Table 2: Average costs (and empirical 95%-confidence interval) over 100 simulated data sets with identical
parameters. Costs are determined exact using Eq. (4). We display the absolute costs for the oracle and the
relative costs w.r.t the oracle for the other approaches. The relative costs are determined by division with
the costs realized under oracle. The average machine profile’s costs are determined by taking the average
over all machine profiles with equal weights.
The lack of sufficient data per profile due to the stratification of the data set, may lead to poor and volatile
The uniform approach pools the data across machine profiles, yet without tailoring the maintenance
policies to each machine profile. This approach alleviates the lack of data, but also leads to an average loss
of 5% compared to the oracle, in this case due to lack of differentiation in the PM policies. Although the
average performance of the uniform approach and the stratified approach is (almost) the same, the spread
on the costs under the uniform approach is much smaller. This is also observed from the smaller 97.5%
quantiles for specific machine profiles compared to the stratified approach.
The pooling approach makes use of all the data, and also differentiates the maintenance policies over
the different machine profiles. This leads to considerably better performance compared to the stratified and
uniform approaches, only having a loss of 0.7% on average with respect to the oracle. The improvement is
not only for the average machine, it also is the case for the different machine profiles. Furthermore, the
spread in performance, as quantified by the 95%-confidence intervals, is considerably smaller compared to
the stratified and uniform approach, both for an average machine and for each individual machine profile.
3.3 How does data pooling overcome the small data problem?
Although the pooling and stratified approaches have a similar goal, i.e. tailoring the maintenance policy
to its machine profile, they use the data set in a different way. While the pooling approach uses all the
data over the different profiles, relying on the assumption of proportional hazards for the failure behaviour
and the assumption of a GLM for the costs, the stratified approach only considers the data per profile
completely disjoint from the others. Splitting the data set per machine profile produces smaller subsets of
data, potentially inducing a small data problem. The consequent underperformance of the stratified approach
is due to the fact that insufficient data may be available per machine profile to estimate the failure behaviour
and costs accurately from the data. Clearly, if sufficient data would be available for each machine profile,
both the stratified and pooling approach converge to the oracle. Yet, the rate at which both approaches
converge is different.
pooling stratified
0 200 400 600
number of machines
relative costs
(a) average machine profile
pooling stratified
0 200 400 600
number of machines
relative costs
(b) worst machine profile
Figure 1: Relative costs for the pooling approach and the stratified approach with respect to the oracle in
function of the number of machines that generate data, averaged over 40 simulated data sets with identical
parameters and with the empirical 90%-confidence interval shaded. Panel (a) Relative costs of the average
machine profile; panel (b) Relative costs of the worst performing machine profile.
Figure 1demonstrates this convergence by displaying the relative costs of the stratified and the pooling
approach with respect to the oracle when we gradually increase the number of machines in the data set,
and thus the number of failure and maintenance records that are used to estimate the failure behaviour and
costs. We start from a data set generated by 10 machines observed during 5 years and consider increases
of 10 machines at a time. These increases correspond to an additional 50 machine years of historical data
(recall that the mean time-to-failure for machine profile x= (0000) is equal to 1.27 years). We report the
average cost performance over 40 simulated data sets with identical parameters and focus on the average
machine and worst performing machine profile, i.e. the machine profile that has the highest relative costs
with respect to the oracle at any given size of the data set. Figure 1shows how both the stratified and
the pooling approach converge to the oracle costs. Yet, the rate of convergence for the pooling is much
higher than the stratified approach. Also the 90%-confidence intervals for the pooling approach are smaller
and shrink faster than the stratified approach. To get insight in the rate of convergence, we consider the
relative costs, averaged over the 40 simulated data sets, of the average profile as a function of the number
of machines, and we look for asuch that,
average relative costs = a
number of machines + 1.
We find fitted values for aequal to 1.715 and 13.341 for the pooling and stratified approach respectively.
The ratio of these values indicates that the pooling approach converges to the oracle over seven and half
times faster than the stratified approach in the number of machines on average. It shows how the pooling
approach requires much less data to obtain adequate performance.
Another downside of the stratified approach is that it cannot prescribe the number of PMs for a machine
profile that is not available in the data set. For instance, if we consider a data set generated by only 10
machines, not all 16 profiles will be represented in the data. This also explains the sharp cost increase of
the worst performing machine profile under the stratified approach when the machine portfolio is small (see
Figure 1, panel (b)). In that case, increasing the number of machines also increases the number of machine
profiles in the data set, and with that also the likelihood of a worse performing profile. This is not the
case under a pooling approach. Even when certain combinations of machine characteristics, i.e. machine
profiles, may not be observed in the data, the pooling approach can still prescribe the preferred number
of PM interventions. This property is essential for an OEM or a service provider when they expand their
maintenance portfolio with machine profiles, for which no data is yet available.
5 10 15 20
number of PMs
condition LHS
(a) Simulation parameters
5 10 15 20
number of PMs
condition LHS
(b) Less heterogeneity
5 10 15 20
number of PMs
condition LHS
(c) Larger Weibull shape γ
Figure 2: Left-hand-side of the optimality condition (dots) and min(cratio(x)) and max(cratio (x)) (dashed
horizontal lines). The available number of PMs determined by min(cratio(x)) and max(cratio (x)) are high-
lighted in light blue.
3.4 When is it valuable to differentiate the PM policy?
Whereas our numerical experiment has shown how our pooling approach is capable to differentiate the
PM policy per machine profile, it is worthwhile to check whether differentiating the maintenance policy is
actually worth the effort compared to adopting a uniform PM policy that is the same for all machines. We
do so by careful analysis of the optimality condition in Proposition 1. Specifically, we consider the equivalent
condition under the assumption of a Weibull failure intensity function,
(αt)γ(n+ 2)γ1(n+ 1)γ1
((n+ 1)(n+ 2))γ1cratio(x),(7)
cratio(x) = cp(x)
cf(x) exp(β·x)
denotes the machine profile specific cost ratio. This ratio provides an indication how many PM interventions
are economic for a machine profile x: a low value of cratio (x) indicates that it is preferred to perform
many PM interventions on this machine profile, as the cost of preventive maintenance is low compared to
the cost of failure, and vice versa for a high value of cr atio(x). The left-hand-side of (7) depends on the
baseline failure intensity, characterized by the Weibull scale αand shape γ, and the contract length t.
Both are independent of the machine profile x. The right-hand-side cratio(x) captures all the effects of the
heterogeneity in the machine portfolio. It depends on the machine profile’s costs of failure via cf(x) and
preventive maintenance via cp(x) and the impact of the profile on the failure behaviour, exp(β·x).
It is insightful to visualize this optimality condition. The dots in Figure 2visualize the left-hand-side of (7)
for each value of n, the number of PM interventions performed during the contract horizon. The minimum
and maximum value of cr atio(x) over the machine profiles, resp. min(cratio(x)) and max(cratio (x)), are
indicated by dashed horizontal lines. These two horizontal lines define the region of the optimal number
of PMs for the machine profiles, defined by x. We highlight these PM policies in light blue. When there
are many light blue dots, the optimal PM policies differ much across the different machine profiles. In the
extreme case where there is only a single point in this region, then the same number of PMs is optimal for
all machine profiles and differentiation of the PM policy is not required.
An extensive analysis of the optimality condition and the resulting number of differentiated PM policies,
reveals that less heterogeneity in the machine portfolio, resulting in a smaller gap between min(cratio(x))
and max(cratio(x)) with the horizontal, dashed lines closer together, leads to less differentiation in the PM
policies over the different machine profiles (see Figure 2b). Also, a larger Weibull shape γof the failure
behaviour (while adapting scale αaccordingly to maintain a constant mean-time-to-failure), makes the left-
hand-side of the optimality condition steeper, i.e. the left-hand-side decreases faster (see Figure 2c). This
reduces the number of different optimal PM policies across the profiles and diminishes the effect of the
heterogeneity of the machine population.
3.5 Model mis-specification vs data-pooling
The data pooling approach requires the specification of relevant terms in (1). In certain contexts, a modeler
may not correctly identify all the terms that should be present in a proportional hazard model. The stratified
approach is immune to such mis-specifications, but requires large amounts of data to obtain accurate esti-
mates. The data-pooling approach is susceptible to such mis-specifications but has the advantage of being
able to pool the data. Below we consider a set-up to study which effect is dominant.
We consider a set-up with only two (binary) covariates, i.e. x1, x2 {0,1}, impacting the time-to-failure.
We simulate the failure times for each machine from following machine-specific failure intensity function
λ(t) = λ0(t) exp βx1+βx2+β
ρx1x2for t[0, tktk1) (k),(8)
where t0= 0 and tn+1 = t, respectively refer to the start and end of the planning horizon. We choose
the same βfor the covariates x1and x2to ensure that the impact of the cross-term (x1x2) is of the same
magnitude as the main effects. The parameter ρcontrols the relative impact of the cross-term. The simulation
of failure and maintenance costs is not changed from the original set-up. We compare the stratified approach
with the data pooling approach but let the data pooling approach fit the following mis-specified model:
λ(t) = λ0(t) exp (βx1+βx2) for t[0, tktk1) (k).(9)
Thus we can interpret ρ1as measuring the amount of model mis-specification of the data pooling approach.
Note that the stratified approach does not suffer from mis-specification since it makes no assumption of the
impact of the covariates on the time-to-failure (nor the failure or maintenance costs). Consequently, the
resulting maintenance policies for the pooling approach will result from the mis-specified model and we will
be able to test the impact of this mis-specification.
In order to assess the impact of the mis-specification, we compare the costs relative to the oracle of both
the pooling approach and the stratified approach for decreasing ρ(increased model mis-specification). The
oracle is of course adapted to consider the influence of the cross-term. The results are displayed in Figure 3.
They show that the effect of data pooling outweighs the effect of model mis-specification in all considered
settings. This is a strong indication, that, while mis-specification is likely to happen, its negative impact is
easily offset by the benefits of data pooling.
4 Conclusion
This paper describes a data-driven approach to optimize the periodic, preventive maintenance policies for a
heterogeneous machine population over a finite time horizon. The heterogeneity of the machine population
is characterized by observable machine characteristics that induce different machine profiles. Our approach
pools the available data of failure and maintenance records over the different machine profiles in order to
learn as best as possible the failure behaviour and the costs of failure and maintenance for each machine
profile. We rely on the assumption of proportional hazards for the failure behaviour and on the assumption of
a gamma GLM for the costs to accomplish this data pooling. In conjunction with the estimates for the failure
behaviour and the costs, our optimality condition for the number of preventive maintenance interventions
delivers tailored maintenance policies for each machine profile. By means of numerical experiments, we
compare our pooling approach with both a stratified approach that splits the data per machine profile and
auniform approach that disregards the machine profiles and prescribes the same uniform PM policy for
all machine profiles. The pooling approach outperforms these benchmarks and additionally has a smaller
spread on its performance. We also show how the pooling approach is more data-efficient than the stratified
approach, even under mild model mis-specification. This means that the pooling approach obtains better
performing maintenance policies for the same amount of data. Finally, we investigate when it is worth
differentiating the PM policy given the heterogeneity in the machine population and the failure intensity,
motivating the approach introduced in this paper.
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pooling stratified
0 50 100 150 200 250
number of machines
relative costs
(a) ρ= 100
pooling stratified
0 50 100 150 200 250
number of machines
relative costs
(b) ρ= 50
pooling stratified
0 50 100 150 200 250
number of machines
relative costs
(c) ρ= 20
pooling stratified
0 50 100 150 200 250
number of machines
relative costs
(d) ρ= 10
pooling stratified
0 50 100 150 200 250
number of machines
relative costs
(e) ρ= 5
pooling stratified
0 50 100 150 200 250
number of machines
relative costs
(f) ρ= 2
Figure 3: Relative costs for the pooling approach and the stratified approach with respect to the oracle in
function of the number of machines that generate data according to the set-up in Section 3.5, averaged over
20 simulated data sets with identical parameters and with the empirical 90%-confidence interval shaded.
A Parameters used to generate data
Time-to-failure α γ β1β2β3β4
0.7 2 0.4 0.3 -0.3 -0.5
PM costs θpβp,0βp,1βp,2βp,3βp,4
15 log(30) 0 0 0 0
Failure costs θfβf,0βf,1βf ,2βf,3βf,4
15 log(300) 0.2
Table 3: Parameter values used to generate a data set of failure and maintenance records, with αand γ
resp. the scale and shape of the Weibull baseline failure intensity function and βithe impact of covariates x,
θpand θfresp. the shape of the gamma distributed failure and preventive maintenance costs, and βp,i , βf,i
their respective machine profile-dependent impact.
B Extract of the generated data set
imachine profile time type costs tiδ
1 1 1 1 0 1 PM 28.26 5 1
1 1 1 1 0 1.91 FAIL 400.33 5 1
1 1 1 1 0 2 PM 29.4 5 1
1 1 1 1 0 3 PM 23.82 5 1
1 1 1 1 0 3.86 FAIL 333.31 5 1
1 1 1 1 0 4 PM 37.74 5 1
1 1 1 1 0 4.93 FAIL 616.39 5 1
1 1 1 1 0 5 END 0 5 0
2 0 1 1 0 1 PM 13.48 5 1
2 0 1 1 0 1.59 FAIL 274.38 5 1
2 0 1 1 0 2 PM 47.39 5 1
2 0 1 1 0 3 PM 25.78 5 1
2 0 1 1 0 3.51 FAIL 254.78 5 1
2 0 1 1 0 4 PM 37.03 5 1
2 0 1 1 0 5 END 0 5 0
3 1 1 0 1 0.98 FAIL 375.79 5 1
3 1 1 0 1 1 PM 29.93 5 1
3 1 1 0 1 2 PM 32.52 5 1
3 1 1 0 1 2.52 FAIL 215.42 5 1
3 1 1 0 1 3 PM 34.29 5 1
3 1 1 0 1 3.71 FAIL 334.9 5 1
3 1 1 0 1 4 PM 24.39 5 1
3 1 1 0 1 4.59 FAIL 212.41 5 1
3 1 1 0 1 5 END 0 5 0
Table 4: Extract of simulated data set with records of three machines.
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EXAMPLES OF SPATIAL POINT PATTERNS INTRODUCTION TO POINT PROCESSES Point Processes on R^d Marked Point Processes and Multivariate Point Processes Unified Framework Space-Time Processes POISSON POINT PROCESSES Basic Properties Further Results Marked Poisson Processes SUMMARY STATISTICS First and Second Order Properties Summary Statistics Nonparametric Estimation Summary Statistics for Multivariate Point Processes Summary Statistics for Marked Point Processes COX PROCESSES Definition and Simple Examples Basic Properties Neyman-Scott Processes as Cox Processes Shot Noise Cox Processes Approximate Simulation of SNCPs Log Gaussian Cox Processes Simulation of Gaussian Fields and LGCPs Multivariate Cox Processes MARKOV POINT PROCESSES Finite Point Processes with a Density Pairwise Interaction Point Processes Markov Point Processes Extensions of Markov Point Processes to R^d Inhomogeneous Markov Point Processes Marked and Multivariate Markov Point Processes METROPOLIS-HASTINGS ALGORITHMS Description of Algorithms Background Material for Markov Chains Convergence Properties of Algorithms SIMULATION-BASED INFERENCE Monte Carlo Methods and Output Analysis Estimation of Ratios of Normalising Constants Approximate Likelihood Inference Using MCMC Monte Carlo Error Distribution of Estimates and Hypothesis Tests Approximate MissingData Likelihoods INFERENCE FOR MARKOV POINT PROCESSES Maximum Likelihood Inference Pseudo Likelihood Bayesian Inference INFERENCE FOR COX PROCESSES Minimum Contrast Estimation Conditional Simulation and Prediction Maximum Likelihood Inference Bayesian Inference BIRTH-DEATH PROCESSES AND PERFECT SIMULATION Spatial Birth-Death Processes Perfect Simulation APPENDICES History, Bibliography, and Software Measure Theoretical Details Moment Measures and Palm Distributions Perfect Simulation of SNCPs Simulation of Gaussian Fields Nearest-Neighbour Markov Point Processes Results for Spatial Birth-Death Processes References Subject Index Notation Index