*Corresponding author. Tel:#44-161-295-4710; fax:# 44-161-295-4947.
E-mail address:d.f.percy@salford.ac.uk (D.F. Percy).
Determining economical maintenance intervals
David F. Percy*, Khairy A.H. Kobbacy
Centre for Operational Research and Applied Statistics, University of Salford, Lancashire M5 4WT, UK
Abstract
Several models have been proposed for scheduling the preventive maintenance (PM) of complex repairable systems in industry. These are often application-speci"c and some make unrealistic assumptions about stationarity of the process and quality of repairs. We investigate two principal types of general model, which have wider applicability. The"rst considers"xed PM intervals and is based on the delayed alternating renewal process. The second is adaptable, allowing variable PM intervals, and is based on proportional hazards or intensities. We describe how Bayesian methods of analysis can improve the decision making process for these models and discuss simulation algorithms for"tting the models to observed data. Finally, we identify some issues that need more research. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Preventive maintenance; Delayed alternating renewal process; Proportional-hazards model; Proportional-intensities model; Bayesian methods; Simulation
1. Introduction
Some of the major expenses incurred by industry relate to the replacements and repairs of manufac-turing machinery in production processes. Condi-tion monitoring [1] and preventive maintenance are the main approaches adopted to reduce these costs. However, most companies devote insu$cient e!ort to modelling their systems and optimising their maintenance strategies, to bene"t fully from the advantages that they o!er. Scarf [2] presents a review of recent mathematical models for main-tenance analysis.
Following on from other scientists, including Watson [3], Handlarski [4] and Dagpunar and
Jack [5], we are investigating the scheduling of preventive maintenance for repairable systems. This paper presents an overview of the models that we have investigated and consider most relevant for practical applications. We also o!er some sugges-tions for future developments, to enhance the pre-dictive power of these models by including concomitant information. In particular, we con-sider the delayed alternating renewal process, for stationary systems, and models based on propor-tional hazards and intensities, for non-stationary systems. These models were applied in the context of preventive maintenance in the two papers by Percy et al. [6] and by Kobbacy et al. [7]. They include standard models for renewals and minimal repairs, as well as age-reduction models as special cases. Makis and Jardine [8] have achieved con-siderable success in scheduling replacements using the proportional hazards model. The most
important features of the current article are its overview of present and future research in this area, with greater emphasis on computational solutions to the decision-making process and less on at-tempts at analytical solutions, which are very un-wieldy.
We also discuss issues relating to the analysis of these models when few data are available about system failures. This situation is very common in practice, particularly when the system is fairly new or the maintenance strategies are reasonably e! ec-tive already. We resolve this problem using the methods of Bayesian decision theory. Our initial ideas and methods were described in the above papers, but we now present a review of this ap-proach and highlight areas that need developing.
The problem that we consider is to schedule preventive maintenance for a repairable system, in order to minimise the expected cost per unit time. We distinguish between systems that do and do not display stationary behaviour, in terms of long-term improvement or deterioration. In both cases, we can optimise the decisions over a "nite or in"nite horizon. Our applications to date relate to the reliability of mechanical equipment such as large valves and pumps in the oil industry.
For a stationary system, analyses using an in" -nite horizon correspond with analyses based on variable horizons of single preventive-maintenance intervals, thereby substantially reducing the com-putational e!ort needed. This simpli"cation does not apply to non-stationary systems. In addition, the former leads to policies for implementing pre-ventive maintenance at regular intervals, whereas the latter forecasts only one period ahead, leading to varying intervals between successive mainten-ance actions.
2. Stationary systems
Consider a repairable system that is subject to failures. We refer to such a system as stationary if there is no long-term improvement or deterioration of its performance. For many applications, the as-sumptions of renewal and minimal repair are too restrictive. We have encountered the need for an alternative scenario that allows for minor
repairs, as follows:
f Corrective maintenance is performed upon
fail-ure, to restore the system to a reasonable operat-ing state.
f Preventive maintenance takes place at regular
intervals, to reset the system to a good operating state.
Corrective maintenance corresponds to minor re-pair work and may involve replacing the damaged components, whereas preventive maintenance cor-responds to minor interventions such as lubricat-ing, cleaning and inspection.
Given this structure, we assume that failure times after corrective operations are independent and identically distributed, as are failure times after preventive maintenance. However, we allow for dif-ferent probability distributions in the two cases and this de"nes the delayed renewal process. Kobbacy et al. [9] investigated this model and derived a for-mula for the probability mass function of the num-ber of failures in a preventive-maintenance interval of given duration, in the case where these two distributions are exponential. For other failure-time distributions, corresponding algebraic for-mulae are very di$cult to obtain. If decisions are based on expectations only, applications of renewal theory yield equations that can be solved numer-ically or graphnumer-ically, see [10}12]. To resolve prob-lems of a more general nature, simulation approaches appear necessary.
This is not an ordinary renewal process, because of the di!erent lifetime distributions following the two types of maintenance. However, the ordinary renewal process could be regarded as a limiting case of our model, if corrective operations were to repair the system to the same state as preventive actions. Of course, all PM might be unnecessary if this were so. Similarly, if corrective actions only restore the system to the state immediately before failure, minimal repairs would result. This is not strictly a special case of our model, but a computer program could easily allow for this assumption if required. However, we believe that minimal repairs are convenient for mathematical modelling but are not always valid in practice.
take place. In many applications involving continu-ous-process industries, the principal costs of main-tenance are not due to parts and labour, but are due to lost production whilst the system is down. Consequently, the downtime must be considered in deciding on cost-e!ective strategies.
This extension results in the delayed alternating renewal process, for which analytical solution is not even feasible for practical purposes. The downtimes following preventive and corrective maintenance can be"xed or random. Since analytical solution of the optimisation problems is not possible and we are adopting a simulation approach here, either of these can be included in the calculations with ease. In the following work, we consider them "xed to avoid confusion. Another bene"t of simulation over numerical solution of the renewal equations is that anomalies are readily catered for, such as switching from CM to PM if the system is in the failed state when PM is due.
Sometimes, a "nite horizon is clearly de"ned. Perhaps a factory or machine is owned on a twenty-year lease. On other occasions, the equip-ment might be retained until cost e$ciencies on a larger scale recommend replacement or scrap-ping. For a pre-determined"nite horizon, we base decisions on simulating the process for the whole horizon. Otherwise, we can simply optimise our strategy over a single preventive-maintenance interval, corresponding with the result for an in" -nite horizon for these stationary models.
De"ne the random variables;and<to be the
lifetimes after preventive maintenance (PM) and corrective maintenance (CM) respectively. Their probability density functions, conditional on known parameters, aref
U(u) andfV(v) respectively. Typically, we consider gamma,
f(t)" ta~1
distributions to achieve the required#exibility (see [13] for details). Note that the exponential distribu-tion is a limiting case of the gamma and Weibull as
aP1. Suppose that the downtimes corresponding to PM and CM arerands, with associated costs candd. These include the costs of parts, labour and downtime.
Assuming an in"nite horizon, we now simulate a PM interval of length t. First, allow r units of downtime for PM and generate a pseudo-random observation u from f
U(u), to represent a typical lifetime following PM. Ifr#u*t, the interval is complete and the total cost incurred isc. However, if r#u(t, we add a CM downtime s. If r#u#s*t, the interval is complete with a cost of c#d. Alternatively, if r#u#s(t, we generate a pseudo-random observationv
1fromfV(v), to rep-resent a typical lifetime following CM. We continue this process, generating CM lifetimesv
1,v2,v3,2 until this interval is complete, and calculate the total cost for the interval in the same manner. Call this total costk
1.
Having completely simulated a PM interval of length t, we repeat this procedure m times and determine the total costs for these simulated inter-vals,k
provides an unbiased estimator for the total cost per PM interval. Hence, we estimate the expected cost per unit time askM/t.
Now the whole simulation must be repeated for di!erent values of t, using an e$cient search algo-rithm, to determine the value of t that minimises this expected cost per unit time. This is the recom-mended PM interval duration. We advocate direct search algorithms for practical implementation, such as golden-section search. For practical pur-poses,t is unlikely to vary continuously and dis-crete values will dominate. Multiples of days, weeks or months provide convenient units of measure-ment, for instance.
We modify the simulation for scenarios involving
a "nite horizon. Instead of simulating one PM
costs of PM and CM over this duration. If we
rede"nek
1as the expected cost over this horizon, then successive replications of this simulated pro-cess generate total costs k
1,k2,2,km as before. This time however, the expected cost per unit time is given by kM/h where kM is the sample mean (see Eq. (4)).
In closing this section, we acknowledge that our model assumptions could be generalised to con-sider di!erent levels of maintenance activity, if these are evident in practice. For example, PM and CM might each be performed as minor or major activ-ities, with corresponding down times. Such possi-bilities are application speci"c and can readily be incorporated as required, by extending the basic simulation program. Indeed, simulation might be the only feasible means of analysis and optimisa-tion in this case.
3. Non-stationary systems
In this section, we allow for systems that display long-term trends, corresponding to improvement or deterioration. We propose two basic models, based on proportional intensities and proportional hazards. These models can also be used for station-ary and non-stationstation-ary systems when concomitant information is available. We discuss these bene"ts later in this section.
For the proportional-intensities model, we con-sider the system's operation as a non-homogeneous Poisson process (NHPP). In many ways, this is the natural formulation. De"ne the random variable N(¹) as the number of system failures by time¹. Then, the NHPP is characterised by conditionally independent increments, corresponding with condi-tionally independent times between failures that occur with intensity
n(¹)"lim t?0
PMN(¹#t)!N(¹)*1DH(¹)N
t (5)
at system age ¹ time units, where H(¹) is the history of the process (see [6]). To investigate this NHPP further, we condition only upon the history at time ¹ to avoid the problems associated with
doubly stochastic processes and obtain
PMN(¹#t)!N(¹)"nDH(¹)N
"MkT(t)Nn
n! expM!kT(t)N (6)
forn"0, 1, 2,2where
k
T(t)"
P
T`tT
n(t) dt (7)
is the mean number of failures in the interval (¹,¹#t). Consequently, as described by Crowder et al. [13], the reliability function for the next failure from time¹is
R
T(t)"PMN(¹#t)!N(¹)"0DH(¹)N "expM!k
T(t)]N, (8)
from which we can determine the lifetime distribu-tion following a particular maintenance acdistribu-tion at time¹:
f
T(t)"!R@T(t)"n(¹#t) expM!kT(t)N. (9) This allows us to simulate the process as before, evaluate expected costs over a"nite horizon, and so deduce the most economical time for the next pre-ventive maintenance. This decision can be made at any speci"c event, such as during PM or CM, or even between events, so long as the intensity func-tion is known.
Common forms for the intensity function of a NHPP include the constant, log-linear and power-law forms:
n
1(t)"a, n2(t)"aect, n3(t)"actc~1. (10) However, these make no allowances for system improvement, or even deterioration, arising from maintenance actions. Hence, we modify these inten-sity functions by introducing a multiplicative fac-tor, so that the intensity function can be expressed as
n(t)"n
0(t) exp(b@z), (11)
where the baseline intensity n
0(t) has one of the forms speci"ed in Eq. (10) and the parameter vector
The vector z contains covariate information
about the system, such as cumulative observations of:
f time since last PM; f time since last CM;
f total number or total downtime of PM's; f total number or total downtime of CM's; f average PM interval duration.
We might consider other covariates for inclusion here, representing the concomitant information mentioned earlier. These could include:
f severity measures of failures;
f condition-monitoring measurements.
Percy et al. [6] considered models of this sort, but reset the time scale of the baseline intensity function to zero upon each PM action. System age was then included amongst the covariates. How-ever, more#exibility is achievable by considering a global time scale for the baseline intensity, as outlined above, or even a mixture of the two. More research is in progress to investigate the possibili-ties (see [14]).
So far, applications of this proportional-inten-sities model have held the covariates at "xed values throughout each PM interval, to avoid com-putational di$culties. This essentially treats all cor-rective maintenance as minimal repair work, an assumption that we earlier claimed is often unreas-onable. To avoid this constraint, we need to con-sider variables that change during a PM interval, such as the cumulative number of failures rather than the total number of failures recorded at the time of the last PM. The computational e!ort re-quired to incorporate temporal covariates is im-mense. Again, further work is underway to resolve this issue.
We now turn our attention to the proportional-hazards model. In principle, this seems inappropri-ate for representing a complex system, because hazards naturally relate to lifetimes of components rather than inter-failure times of processes. That we cannot physically justify this model as readily as the proportional-intensities model does not invali-date its use in this context though. Makis and Jardine [8] have used the proportional hazards model successfully for replacement decisions and
preliminary analyses by Kobbacy et al. [7] in the context of preventive maintenance are quite prom-ising.
Since hazard relates to a speci"c time until fail-ure, whereas intensity relates to the whole stochas-tic process, we need to adopt di!erent hazard functions after PM and CM. However, these might be of the same functional forms as the intensity function:
i(t)"i0(t) exp(c@x) (12) for hazards following PM and
j(t)"j0(t) exp(d@y) (13)
for hazards following CM, wheretnow measures time since the most recent event. The baseline in-tensities in Eq. (10) can be used for the baseline hazards in Eqs. (12) and (13), corresponding to the exponential, Gumbel and Weibull distributions re-spectively. The covariates that might be contained in the vectorsxandyare as for the
proportional-intensities model: we again have computational problems with temporal covariates. The vectors
canddcontain the regression coe$cients. To avoid referring separately to the hazard func-tionsi(t) andj(t), consider a general hazard func-tion h(t). For the purposes of simulation, the cumulative distribution function can be determined as
F(t)"1!exp
G
!P
t 0h(t) dt
H
, (14)from which the probability density function is
f(t)"F@(t)"h(t)exp
G
!P
t 0h(t) dt
H
. (15)Preliminary analyses indicate that this propor-tional-hazards model is#exible and avoids some of the problems associated with the proportional-intensities model, despite a clearer mathematical and physical justi"cation for the latter.
4. Simulation and Bayesian analysis
di$cult and we recommended simulation of the process for practical purposes. For accurate results, the computing power required is substantial and so e$cient algorithms are required for generating pseudo-random numbers (PRNs) from given prob-ability distributions.
For our purposes, we identi"ed a small number of distributions that arise frequently in application. Predominantly, we consider gamma, Weibull and log-normal lifetimes. The Fortran NAG library o!ers computer program routines for generating PRNs from these routines directly, and we make use of these facilities for our research. Typically, we have found that simulation of a million PM inter-vals generates expected costs to 3 or 4 signi"cant
"gures of accuracy within several minutes on
a Pentium computer.
We also need to generate bivariate random num-bers. In most cases, this can be achieved by writing a joint density as the product of a marginal and a conditional density. Then, a PRN is generated from the marginal density, substituted into the con-ditional density and a PRN is generated from the latter, giving the required bivariate PRN.
So far, we have assumed that all of the para-meters are known. For all of the above models however, the parameters are generally unknown. In the context of stationary models, these include parameters of the lifetime distributions following PM and CM. For non-stationary models, these include parameters in the baseline hazards or intensities and regression coe$cients in the multi-plicative factors (see Eqs. (11)}(13)).
The usual approach for coping with unknown parameters is to estimate them from a random sample of data and then assume that the parameters are equal to these estimates. For the application that we are considering, our data con-sist of independent lifetimes after PM and CM, some of which are right-censored when PM inter-vened before failure.
However, this approach is fundamentally#awed because the parameters are not equal to their max-imum likelihood estimates. Nevertheless, the tech-nique of parameter estimation is so common in practice because it is relatively easy to adopt and usually gives accurate results. The main situation where accuracy is compromised is when few data
are available, from which to estimate the para-meters. In these circumstances, inference can be quite wrong (see [15}17] for evidence of this). Many applications in the areas of reliability and maintenance are notorious for their lack of data (see [18]).
Martz and Waller [19] adopted a Bayesian approach to reliability analysis, as a way of avoid-ing errors through estimatavoid-ing parameters. We have applied the method to some of the models de-scribed earlier, making reasonable progress to-wards the analytical solution of the di$cult integrals that arise [20]. Simulation-based opti-misation is preferred though, because of its wide-spread applicability and ease of implementation, despite the large computational e!ort required.
Before proceeding, we must avoid confusion by always indicating conditional dependency upon unknown parameters where appropriate. That is, we write f(tDH) for the model's sampling density
instead of f(t), where H is the set of unknown
parameters. We also need similar notation for the cumulative distribution, reliability, hazard and intensity functions. The essence of the Bayesian approach is to specify a prior densityg(H) on the unknown parameters. We can then determine all desired forms of inference and decision by routine applications of probability and calculus results, generating random lifetimes from the posterior pre-dictive density
f(tDD)"
P
H
f(tDH)g(HDD) dH, (16)
where D is the set of observed data, rather than from the assumed sampling density f(tDH) evalu-ated at the maximum likelihood estimate HK . The second term in the integrand of Eq. (16) represents the posterior density ofHgivenDand is given by
g(HDD)J¸(H;D)g(H), (17)
where ¸(H;D) is the same likelihood function as would be required for determining estimates, if the classical approximation were adopted instead.
In order to generate pseudo-random lifetimes from the posterior predictive density in Eq. (16), we
"rst generate pseudo-random values for the
Eq. (17). Then, we use these values to generate pseudo-random lifetimes from the sampling density f(tDH). All that remains, mathematically, is the substantial problem of determining suitable forms of prior distribution for the models described earlier. This issue was addressed by Singpurwalla and Percy [21] and is the subject of ongoing research.
5. Conclusions
In this paper, we have reviewed progress to date on developing a suite of models and simulation programs for scheduling preventive maintenance of complex repairable systems. We envisage the de-layed alternating renewal process for systems that exhibit stationarity, in which case optimisation is computed over a"nite or variable horizon. Simpler models based on replacements and minor repairs will apply in some situations, whilst the simulation approach proposed also enables more general models to be used.
For non-stationary systems, we propose the pro-portional-hazards or proportional-intensities mod-els. The assumptions for these two models di!er slightly because the baseline time scale is local for the former and global for the latter. Further research is under way to help decide between them. Optimisation is now over a "nite horizon only, as analyses over in"nite horizons do not simplify and cannot be computed for these models. E$cient dynamic programming algorithms are needed here. We have also made a good start at allowing for unknown parameters, using a Bayesian approach to the analysis. This will be particularly bene"cial when few failure times have been observed, perhaps when a relatively new process is involved, and when the model needs regular updating in line with sys-tem changes or monitoring improvements.
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