*Corresponding author. Tel.: 745-5000; fax: 44-161-745-5559.
Re
"
ning the delay-time-based PM inspection model with
non-negligible system downtime estimates of the
expected number of failures
A.H. Christer
!,",
*, C. Lee
"
!Center for OR and Applied Statistics, University of Salford, Salford M5 4WT, UK
"Eindhoven University of Technology, Faculty of Technology Management/Section Product and Process Quality, P.O. Box 513,
5600 MB Eindhoven, The Netherlands
Abstract
In this paper a re"nement to the delay-time-based PM model is presented to account for the downtime incurred at failures when estimating the expected number of failures over a PM inspection period. Extensions to PM models have been made in the context of downtime modelling. Previously, in downtime modelling using delay time, it has been generally assumed that the downtime of failures is small compared with a PM cycle length and, therefore, assuming that defects continue to arise uninterrupted during a downtime period has negligible consequences. However, in some cases, it is possible that the cumulative downtime of failure repairs is not su$ciently small. The paper presents revised models for perfect and non-perfect homogeneous processes, and for a perfect non-homogeneous process. Numerical examples are provided to highlight the di!erences. The revised downtime model is a more sensitive model with which to determine the actual downtime or cost. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Downtime model; Delay time; Preventive maintenance; Modelling
1. Introduction
Delay-time-based modelling of industrial main-tenance inspection problems has been developed and applied within a variety of case studies over the past 10 years, as reported in the DTM review [1]. In most cases, the objective has been to reduce
plant downtime. This generally entails"nding the
appropriate PM or inspection interval, which is the type of modelling problem DTM was developed to address. In formulating delay-time-based
mainten-ance models, an approximation is generally made to aid in estimating the number of failures expected over a PM or inspection period. This paper exam-ines the consequence of this approximation and
presents the modi"ed models relaxing the
approxi-mating assumption.
The delay time concept regards the failure
pro-cess as a two-stage propro-cess with the"rst stage being
when a detectable defect arises, and the second stage when the defect leads to a failure. The time
lapse from the time of the"rst possible identi"
ca-tion of a defect to the point where a repair is essential is called the delay time. If an inspection is carried out during the delay time period of a defect,
the defect may be identi"ed and removed.
Fig. 1. In the development and application of DTM, it
has been generally assumed that the downtime due to failures over a PM inspection cycle is small
compared to the cycle length T. This assumption
permits the expected number of failures arising over a maintenance cycle to be readily ap-proximated. There has, so far, only been one case when this simplifying assumption has not been adopted [2]. Here we establish the extension to the standard DTM that applies when the total down-time due to failure over a PM period may not be
assumed su$ciently small compared with the cycle
periodT. In this case, the impact of downtime due
to failure will impact upon the expected number of failures, and through this upon the model. There are two classes of models that concern us, complex plant models and component tracking models. First, we address the more important complex plant case.
2. Complex plant modelling assumptions
We are concerned with modelling the inspection decision process of a system in which independent
defects having a delay time h may arise when in
operation. Here we consider the general case of an inspection policy which may be characterized by the following assumptions.
(1) PM inspection is undertaken every¹calendar
time units, requires an expectedd
1 time units,
and all identi"ed defects are repaired.
(2) Inspection is perfect in that, if a defect is pres-ent at the time of inspection, it will be idpres-enti-
identi-"ed.
(3) Defects are assumed to arise within the system
at a ratej(u) at operating timeusince the last
PM period.
(4) Failures arising during operating time are
identi"ed and repaired immediately with
ex-pected downtimed
&independent of the defect's delay time.
(5) Defects are assumed to only arise, deteriorate, and lead to failures whilst the system is operat-ing.
(6) The delay time h of a defect is independent
of its time origin and has pdf f(z) and cdf
F(z).
(7) In the event of plant stoppage due to a delay elsewhere, any unexpired delay time of a fault will remain frozen until the plant re-starts.
3. Inspection models
3.1. Basic inspection models
Between inspections, each failure is repaired in
timed
& and the plant continues in operation until halted because of another failure or the PM
inspec-tion at time¹ (see Fig. 1). Let NH
&(¹) denote the expected number of failures over the calendar
period¹.
Let qand N
&(q) denote the expected operating
time over (0,¹) and the expected number of failures
over operating timeq. Since defects can only arise
or deteriorate when the plant is in operation, the
number of failuresNH&(¹) arising over calendar time
(0,¹) is given by
Clearly, this requires¹'d
&N&(q), which may be
considered as a bound ond
&, or upon the expected
number of failures over (0,¹), which is bounded by
N
&(¹). If this condition were not valid, either the
periodTbeing considered was inappropriate, or the
problem was of an insurance nature when the con-sequence of failure was potentially catastrophic.
A di!erent risk-orientated approach to modelling
would apply in this case. For the rest of this paper,
we assume ¹'d
&N&(q). Eq. (2) represents the transformation between the expected operating
If the objective is to select ¹ to minimize the overall downtime, the objective function to
deter-mine the inspection period ¹ to minimize the
expected downtime per unit time over the PM
inspection period,D(¹), is given by
D(¹)"
G
d&N&(q)#d1&;¹, we may for the purpose of modelling the expected number of failures adopt the
conve-nient practice and assume d
&"0, in which case
Of course, having made this approximation, once
the optimal¹is known it is necessary to check that
d
&N&(¹) is also small compared to ¹.
In the steady-state case of homogenous Poisson
arrival of defects, that isj(u)"ja constant, Eq. (4)
forD(¹) reduces to the well-known form given by
Christer and Waller [3], namely
D(¹)"j¹b(¹)d&#d1
b(¹) is the probability that a defect arising at
ran-dom with (0, ¹) will result in a failure.
We have, therefore, that allowing for non-zero
downtime in determining NH&(¹), and permitting
j"j(u), transforms the basic downtime per unit
time equation (5) into the equation set (3). To solve
Eq. (3) for optimal¹, the simplest procedure is to
assume a q value, and calculate the associated¹
period andN
&(q) using Eq. (3). Repetition over an
appropriate mesh of q values will establish
parametrically the appropriate¹value or range.
This formulation has, of course, assumed the rate
of defects,j(u), to be a function of operating time
since the last PM inspection. This implies that the PM has a form of renewal property for the plant. Should this not be valid, and if the failure rate is
a function of actual calendar time, then a di!erent
formulation is required in which inspection period may became variable, and the PM inspection prob-lem becomes in part a replacement probprob-lem [4,5].
3.2. Non-perfect inspection case
Not all inspections are perfect. When a defect at inspection may be detected with probability
b, 0)b)1, in the case of non-negligible
down-time for failure estimation, the delay down-time model modi"cation is relatively simply obtained when
j"constant. In the steady state corresponding to
j"constant, let the probability that a defect will
result in a failure over operating timeqbeb(q). In
the case of regular inspection on period ¹, by
de"ning operating time appropriately we have from Christer and Waller [2] that
b(q)"1!
GP
qWith this revised formulation ofN
&(q) to re#ect non-perfect inspection, the non-zero downtime ad-justed downtime per unit time model of Eq. (3) still applies, and as before is readily calculable over
a meshq.
3.3. The component tracking case
Allowing for non-negligible downtime for failure repair is relatively simple in the case of component tracking models. Here there is essentially only one failure mode being considered, and therefore only one defect at most may be present at any given time
point. Such a modelling has signi"cance in
reliabil-ity centred maintenance decision-making model-ling. Assume for now that a PM inspection returns the components to a post inspected standard condition.
Letg(u) be the pdf of the initial timeuafter an
last inspection to failure in the absence of further
inspection intervention isr(t) where
r(t)"
P
t0
g(u)f(t!u) du. (9)
If inspection is undertaken at age¹, and repair or
replacement to post inspected state initiated im-mediately upon failure, we have an age-based re-placement process in which the expected downtime per unit time is given by
D(¹)"
G
d&F(¹)#d1(1!F(¹)(¹#d
1)(1!F(¹))#d&:T0tr(t) dt
H
. (10)
In this case, the calendar time¹and the
operat-ing timeqare identical.
The less trivial variant of this model is the block replacement case where a PM inspection upgrading
plant to a&post inspected'condition is undertaken
every ¹ calendar time units, with failure repairs
being undertaken with downtimed
& as they arise.
In this case, ifqis the actual operating time, then
the expected number of failures is given by N
&(q)
The following examples demonstrate the in#
u-ence of the non-negligible downtime assumption in determining the expected downtime due to failures in delay time modelling. Interest is restricted to the more common case of a complex plant. The key
parameters are j,b,f(h),d
1 andd&.
For prototype PM models, we assume that the pdf of delay time is an exponential distribution,
f(h)"ae~ahwitha"0.05 per hour, giving an
aver-age delay time of 20 hours, and the averaver-age defect
arrival frequency has been taken asj"0.2 defects
per hour. The mean downtime for a failure and PM
ared
&"0.8 hours andd1"0.3 hours respectively. The objective function for the downtime has the form
D(¹)"
Expected number of failures in a cycle(0,¹)]d
&#d1 ¹#d
1
.
(13)
Di!erent models di!er in the form of the expected
number of failures.
4.1. Basic model
Neglecting the in#uence of downtime in
estima-ting the expected number of failures over a cycle,
when the pdf of delay time is f(h)"ae~ah, the
probability b(¹) that a defect arises as a failure,
Eq. (6), is
Therefore, the expected total downtime per unit
time,D(¹) is
4.2. Non-perfect inspection case
Here, the probability of a defect present during
an inspection being detected,b, is taken asb"0.5.
For the unadjusted expected total downtime
per unit time imperfect inspection model, D(¹)
is given by Eq. (5), with b(¹) being given by
Eq. (16).
4.3. Non-homogeneous defect arrival rate case
We consider here the non-homogeneous case of
Eq. (1) where defect arrival frequency at timeuafter
a perfect inspection is given for demonstration purpose by
j(u)"0.2!0.06e~0.2u. (17)
Then, the expected number of failures arising in
(0,¹), N
and the expected total downtime per unit time,
D(¹), in the non-homogeneous defect arrival rate
case is given by Eq. (4).
4.4. Revised basic PM model for non-negligible downtime in estimating the expected number of failures
This model also assumes that the expected num-ber of defects arising in the PM interval with actual
operating periodqisjq, as in the basic model. We
have for the PM interval of length¹, the number of
failures over an actual operating time (0,q) from
Eq. (6). Therefore, assuming perfect inspection, we
have forb(q) the expected total downtime per unit time for the
revised basic PM model is given by
D(¹)"
4.5. Revised non-perfect inspection case
In a similar way,b(q) for the imperfect PM model
case from Eq. (16) is
It also follows from the form of Eq. (3) that the expected total downtime per unit time for the revised non-perfect PM model is given by
D(¹)"jqb(q)d&#d1
¹#d
1
. (22)
4.6. Revised non-homogeneous defect arrival case
For the re"nement of the non-homogeneous
defect arrival case PM model, we also assume that the instantaneous rate of defect occurrence at time
u after PM is not constant but is given by
j(u)"(0.2!0.06e~0.2u), and a defect arising within
the period (0,s) has a delay time in the interval
(h,h#dh), with probabilityf(h) dh. Therefore, the
expected number of failures arising over actual
op-erating time (0,q) in the non-homogeneous defect
arrival rate case is given by Eq. (1), that is,N
&(q) is
Therefore, the expected total downtime for the revised downtime model is given by Eq. (3) where N
Fig. 3. Expected downtime for non-perfect inspection model and revising model. Fig. 2. Expected downtime for basic model and revised model. Now we shall consider the numerical
conse-quences to the expected downtime for the various
models in terms of calendar time, ¹, and actual
operating time, q. The results for the models
outlined above are shown in Figs. 2}4 and Tables
1}3. First, consider the basic model. It can be seen
from Fig. 2 that if the mean failure downtime,d
Fig. 4. Expected downtime for non-Homogeneous case model and revised model.
Table 1
Optimal downtime results of basic model
Mean downtime d
&"0.3 d&"0.5 d&"0.8
Non-re"ned model Expected unit downtime 0.0365 0.0484 0.0622
Optimum inspection period 19 13 10
Re"ned model Expected unit downtime 0.0357 0.0471 0.0601
Optimum inspection period 20 14 11
Table 2
Optimal results for downtime of non-perfect inspection case model
Mean downtime d
&"0.3 d&"0.5 d&"0.8
Non-re"ned model Expected unit downtime 0.051 0.0713 0.0946
Optimum inspection period 19 11 7
Re"ned model Expected unit downtime 0.0489 0.0681 0.0891
Optimum inspection period 21 13 9
non-re"ned calendar time, ¹, is 10 hours and
downtime per unit time is 0.0622 hours, whilst the
optimal point for the re"ned model based on the
actual time,q, is 11 hours and downtime per unit
time is 0.0601 hours. That is, the expected total downtime for the basic model based on the
calen-dar time,¹, is slightly higher, as would be expected,
since the model overestimates the number of
fail-ures. If the mean failure downtimes,d
&, are 0.3 and 0.5, the optimal point and expected total downtime
curves based on the calendar time,¹and operating
Table 3
Optimal results for downtime of non-homogeneous defect arrival case model
Mean downtime d
&"0.3 d&"0.5 d&"0.8
Non-re"ned model Expected unit downtime 0.0342 0.0447 0.0567
Optimum inspection period 19 13 10
Re"ned model Expected unit downtime 0.0335 0.0436 0.0552
Optimum inspection period 20 14 11
Clearly and as expected, as d
& increases, D(¹) increases and the optimal PM interval decreases.
For the parameters chosen, the di!erence in model
prediction attributable to the modelling re"nement
is evident, but not excessive. The approximation
that d
&"0 for the purpose of estimating N&(q) would appear valid here.
We now consider the imperfect inspection case. In
Fig. 3 plots forD(¹) are shown for the imperfect PM
case based on the calendar time¹and actual
oper-ating timeq. It can be seen that a larger change in the
optimum values of D(¹) is evident with the re"
ne-ment. The optimal interval of the prototype model
based on calendar time¹, in the case of imperfect
inspection, is 7 hours and the expected total
down-time is 0.0946 hours if the mean failure downdown-time,d
&, is 0.8 hours. This extends to a PM period of 9 hours with an expected downtime per unit time of 0.0891
for the more accurate re"ned model.
This result shows that there is a greater di!erence
between the basic and downtime revised model in the non-perfect inspection case compared to the perfect inspection case. The explanation rests in the fact that the model for imperfect inspections has a higher frequency of failures than the basic model. It can also be seen from Table 2 and Fig. 3 that
a similar di!erence between the optimum values of
D(¹) resulting from calendar time and the re"ned
actual operating time models is evident, with the mean downtimes for failures of 0.3 and 0.5 hours.
Again, Fig. 4 shows the expected total downtime of the model for a non- homogeneous defect arrival rate over the calendar time and actual operating time. This case also shows the optimal interval and the expected total downtime for the calendar-time-based model, namely, 10 hours and 0.0567 hours,
respectively, if the mean failure downtime, d
&, is 0.8 hours. The optimal interval and the expected
total downtime for the revised model are 11 hours and 0.0552 hours. Also, the model for non-homo-geneous defect arrival rate shows that the expected downtime is slightly less in the revised downtime formulation case, as would be expected. As the process experiences a high frequency of failures,
a greater di!erence would be expected between
expected downtime per unit time for the two mod-els based upon calendar time and actual operating time for a non-homogeneous defect arrival rate.
This di!erence is evident. Here, Table 3 shows the
optimum values ofD(¹) for the di!erent downtimes
between calendar time and the re"ned actual
oper-ating time model resulting from updoper-atingb(¹).
Although the percentage savings in total down-time in the cases considered above is small, never
much more than 5%, the "nancial consequences,
which depend upon the value of such saving in downtime may be very attractive. Therefore, the revised form of PM model can provide greater accuracy for good decision-making of maintenance activities.
5. Conclusions
Delay time analysis has already proved useful in the rudimentary applications made so far in which approximate models have been used in estimating the expected number of failures over a PM/inspec-tion period. Here the characteristic models have
been extended to re"ne the process of estimating
downtime in formulating defect arrival patterns essentially increases the period in which defects are assumed to arise. The implication is that the revised downtime model would be a more sensitive model with which to determine the actual downtime or
cost. The di!erence is more signi"cant as the
qual-ity of inspection decreases.
References
[1] A.H. Christer, Developments in delay time analysis for modelling plant maintenance, Journal of Operational Re-search Society 50 (11) (1999) 1120}1137.
[2] J.B. Chilcott, A.H. Christer, Modelling of condition-based maintenance at the coal-face, International Journal of Pro-duction Economics 22 (1991) 1}11.
[3] A.H. Christer, W.M. Waller, Delay time models of indus-trial maintenance problems, Journal of Operational Re-search Society 35 (1984) 401}406.
[4] A.H. Christer, W. Wang, A delay-time based maintenance model of a multi-component system, IMA Journal of Mathematics Application in Manufacturing and Industry, 6 (2) (1995) 205}222.
