Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol67.Issue1.Aug2000:
Int. J. Production Economics 67 (2000) 77}85
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$ciently small. The paper presents revised models for
perfect and non-perfect homogeneous processes, and for a perfect non-homogeneous process. Numerical examples are
provided to highlight the di!erences. The revised downtime model is a more sensitive model with which to determine the
actual downtime or cost. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Downtime model; Delay time; Preventive maintenance; Modelling
1. Introduction
Delay-time-based modelling of industrial maintenance inspection problems has been developed
and applied within a variety of case studies over the
past 10 years, as reported in the DTM review [1].
In most cases, the objective has been to reduce
plant downtime. This generally entails "nding the
appropriate PM or inspection interval, which is the
type of modelling problem DTM was developed to
address. In formulating delay-time-based mainten-
* Corresponding author. Tel.: 44-161-745-5000; fax: 44-161745-5559.
ance models, an approximation is generally made
to aid in estimating the number of failures expected
over a PM or inspection period. This paper examines the consequence of this approximation and
presents the modi"ed models relaxing the approximating assumption.
The delay time concept regards the failure process as a two-stage process with the "rst stage being
when a detectable defect arises, and the second
stage when the defect leads to a failure. The time
lapse from the time of the "rst possible identi"cation of a defect to the point where a repair is
essential is called the delay time. If an inspection is
carried out during the delay time period of a defect,
the defect may be identi"ed and removed.
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 1 1 - 6
78
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
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 approximated. 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 downtime due to failure over a PM period may not be
assumed su$ciently small compared with the cycle
period T. 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.
(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
time d and the plant continues in operation until
&
halted because of another failure or the PM inspection at time ¹ (see Fig. 1). Let NH(¹) denote the
&
expected number of failures over the calendar
period ¹.
Let q and N (q) denote the expected operating
&
time over (0, ¹) and the expected number of failures
over operating time q. Since defects can only arise
or deteriorate when the plant is in operation, the
number of failures NH(¹) arising over calendar time
&
(0, ¹) is given by
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 every ¹ calendar
time units, requires an expected d time units,
1
and all identi"ed defects are repaired.
(2) Inspection is perfect in that, if a defect is present at the time of inspection, it will be identi"ed.
(3) Defects are assumed to arise within the system
at a rate j(u) at operating time u since the last
PM period.
(4) Failures arising during operating time are
identi"ed and repaired immediately with expected downtime d independent of the defect's
&
delay time.
(5) Defects are assumed to only arise, deteriorate,
and lead to failures whilst the system is operating.
(6) The delay time h of a defect is independent
of its time origin and has pdf f (z) and cdf
F(z).
P
NH(¹)"N (q)"
&
&
q
j(u)F(q!u) du,
(1)
u/0
where
q"¹!d N (q).
(2)
& &
Clearly, this requires ¹'d N (q), which may be
& &
considered as a bound on d , or upon the expected
&
number of failures over (0, ¹), which is bounded by
N (¹). If this condition were not valid, either the
&
period T being considered was inappropriate, or the
problem was of an insurance nature when the consequence 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
time q and calendar time ¹.
Fig. 1.
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
If the objective is to select ¹ to minimize the
overall downtime, the objective function to determine the inspection period ¹ to minimize the
expected downtime per unit time over the PM
inspection period, D(¹), is given by
G
H
d N (q)#d
& &
1 ,
(3)
¹#d
1
where q"¹!d N (q) and N (q)":q j(u)F(q!u) du.
0
& &
&
If d ;¹, we may for the purpose of modelling
&
the expected number of failures adopt the convenient practice and assume d "0, in which case
&
Eq. (3) for D(¹) reduces to
D(¹)"
H
G
d N (¹)#d
& &
1 ,
¹#d
1
D(¹)"
(4)
where
P
T
j(u)F(¹!u) du.
0
Of course, having made this approximation, once
the optimal ¹ is 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 is j(u)"j a constant, Eq. (4)
for D(¹) reduces to the well-known form given by
Christer and Waller [3], namely
N (¹)"
&
j¹b(¹)d #d
&
1,
D(¹)"
¹#d
1
where
PA
B
(5)
T ¹!h
f (h) dh.
(6)
¹
0
b(¹) is the probability that a defect arising at random 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 optimal ¹, the simplest procedure is to
assume a q value, and calculate the associated ¹
period and N (q) using Eq. (3). Repetition over an
&
appropriate mesh of q values will establish
parametrically the appropriate ¹ value or range.
This formulation has, of course, assumed the rate
of defects, j(u), to be a function of operating time
b(¹)"
79
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 problem becomes in part a replacement problem [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 downtime for failure estimation, the delay 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 time q be b(q). In
the case of regular inspection on period ¹, by
de"ning operating time appropriately we have from
Christer and Waller [2] that
GP
b(q)"1!
H
= b
+ (1!b)n(1!F(nq!u)) du ,
q
u/0 n/1
(7)
q
where N (q)"jqb(q) and ¹"q#d N (q).
&
& &
With this revised formulation of N (q) to re#ect
&
non-perfect inspection, the non-zero downtime adjusted downtime per unit time model of Eq. (3) still
applies, and as before is readily calculable over
a mesh q.
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 reliability centred maintenance decision-making modelling. Assume for now that a PM inspection returns
the components to a post inspected standard
condition.
Let g(u) be the pdf of the initial time u after an
inspection. We have that the pdf of time t from the
80
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
last inspection to failure in the absence of further
inspection intervention is r(t) where
P
t
g(u) f (t!u) du.
(9)
0
If inspection is undertaken at age ¹, and repair or
replacement to post inspected state initiated immediately upon failure, we have an age-based replacement process in which the expected downtime
per unit time is given by
r(t)"
H
G
d F(¹)#d (1!F(¹)
&
1
.
(10)
(¹#d )(1!F(¹))#d :Ttr(t) dt
1
& 0
In this case, the calendar time ¹ and the operating time q are 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 downtime d as they arise.
&
In this case, if q is the actual operating time, then
the expected number of failures is given by N (q)
&
where
D(¹)"
P
N (q)"
&
q
(1#N (q!u))r(u) du.
(11)
&
u/0
Since ¹"q#N (q)d , the expected downtime per
&
&
unit time model D(¹) is given by
G
H
N (q)d #d
&
1 ,
(12)
D(¹)" &
¹#d
1
where ¹"q#N (q)d , and N (q) is given by
&
&
&
Eq. (11).
4. Numerical examples
The following examples demonstrate the in#uence 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 and d .
1
&
For prototype PM models, we assume that the
pdf of delay time is an exponential distribution,
f (h)"ae~ah with a"0.05 per hour, giving an average delay time of 20 hours, and the average defect
arrival frequency has been taken as j"0.2 defects
per hour. The mean downtime for a failure and PM
are d "0.8 hours and d "0.3 hours respectively.
&
1
The objective function for the downtime has the
form
D(¹)"
Expected number of failures in a cycle(0, ¹)]d #d
&
1.
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 estimating 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
P
T¹!h
1
ae~ah dh"1#
(e~aT!1). (14)
a¹
¹
0
Therefore, the expected total downtime per unit
time, D(¹) is
b(¹)"
A
B
1
j¹ 1#
(e~aT!1) d #d
&
1
a¹
D(¹)"
.
¹#d
1
(15)
4.2. Non-perfect inspection case
It is shown in Eq. (7), that in the case b)1,
P
T = b
+ (1!b)n~1M1!(1!e~a(nT~y))N dy
¹
0 n/1
b =
"1!
+ M(1!b)n~1e~anT(e~aT!1)N
a¹
n/1
=
b
(!e~aT!1) + M(1!b)n~1e~anTN
"1!
a¹
n/1
b(1!e~aT)
"1!
a¹(1!(1!b)e~aT)
b(¹)"1!
0.5(1!e~aT)
"1!
.
a¹(1!(1!0.5)e~aT)
(16)
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
Here, the probability of a defect present during
an inspection being detected, b, is taken as b"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).
81
revised basic PM model is given by
B
A
1
jq 1# (e~aq!1) d #d
&
1
aq
,
D(¹)"
¹#d
1
(20)
where q"¹!jqb(q)d .
&
4.3. Non-homogeneous defect arrival rate case
4.5. Revised non-perfect inspection case
We consider here the non-homogeneous case of
Eq. (1) where defect arrival frequency at time u after
a perfect inspection is given for demonstration
purpose by
In a similar way, b(q) for the imperfect PM model
case from Eq. (16) is
j(u)"0.2!0.06e~0.2u.
b(q)"1!
(17)
Then, the expected number of failures arising in
(0, ¹), N (¹), is
&
P
P
N (¹)"
&
"
T
j(u)F(¹!y) dy
0
T
(0.2!0.06e~0.2y)M1!e~0.05(T~y)N dy,
0
(18)
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 number of defects arising in the PM interval with actual
operating period q is jq, as in the basic model. We
have for the PM interval of length ¹, the number of
failures over an actual operating time (0, q) from
Eq. (6). Therefore, assuming perfect inspection, we
have for b(q)
P
q q!h
1
ae~ah dh"1# (e~aq!1),
aq
q
0
P
q = b
+ (1!b)n~1M1!(1!e~a(nq~y))N dy
q
0 n/1
b(1!e~aq)
"1!
.
aq(1!(1!b)e~aq)
(21)
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
jqb(q)d #d
&
1.
D(¹)"
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 probability f (h) dh. Therefore, the
expected number of failures arising over actual operating time (0, q) in the non-homogeneous defect
arrival rate case is given by Eq. (1), that is, N (q) is
&
given by
P
N (q)"
&
q
0
(0.2!0.06e~0.2y)M1!e~0.05(q~y)N dy.
(19)
(23)
where q"¹!jq b(q)d . It follows from Eq. (2) that
&
the expected total downtime per unit time for the
Therefore, the expected total downtime for the
revised downtime model is given by Eq. (3) where
N (q) is given by Eq. (23).
&
b(q)"
82
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
Now we shall consider the numerical consequences 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 ,
&
is 0.8 hours, the optimal point based on the
Fig. 2. Expected downtime for basic model and revised model.
Fig. 3. Expected downtime for non-perfect inspection model and revising model.
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
83
Fig. 4. Expected downtime for non-Homogeneous case model and revised model.
Table 1
Optimal downtime results of basic model
Non-re"ned model
Re"ned model
Mean downtime
d "0.3
&
d "0.5
&
d "0.8
&
Expected unit downtime
Optimum inspection period
Expected unit downtime
Optimum inspection period
0.0365
19
0.0357
20
0.0484
13
0.0471
14
0.0622
10
0.0601
11
Mean downtime
d "0.3
&
d "0.5
&
d "0.8
&
Expected unit downtime
Optimum inspection period
Expected unit downtime
Optimum inspection period
0.051
19
0.0489
21
0.0713
11
0.0681
13
0.0946
7
0.0891
9
Table 2
Optimal results for downtime of non-perfect inspection case model
Non-re"ned model
Re"ned model
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 failures. 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, ¹ and operating
time, q, are shown in Table 1 and Fig. 2.
84
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
Table 3
Optimal results for downtime of non-homogeneous defect arrival case model
Non-re"ned model
Re"ned model
Mean downtime
d "0.3
&
d "0.5
&
d "0.8
&
Expected unit downtime
Optimum inspection period
Expected unit downtime
Optimum inspection period
0.0342
19
0.0335
20
0.0447
13
0.0436
14
0.0567
10
0.0552
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 for D(¹) are shown for the imperfect PM
case based on the calendar time ¹ and actual operating time q. It can be seen that a larger change in the
optimum values of D(¹) is evident with the re"nement. The optimal interval of the prototype model
based on calendar time ¹, in the case of imperfect
inspection, is 7 hours and the expected total downtime is 0.0946 hours if the mean failure downtime, 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-timebased 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-homogeneous 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 models 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 of D(¹) for the di!erent downtimes
between calendar time and the re"ned actual operating time model resulting from updating b(¹).
Although the percentage savings in total downtime 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/inspection period. Here the characteristic models have
been extended to re"ne the process of estimating
the expected number of failures over a PM period.
From the revised PM model, it is evident that the
expected downtime of failures over PM intervals
may be overestimated by not allowing for downtime in modelling the expected number of failures.
This result is not surprising, since ignoring
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
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 quality of inspection decreases.
References
[1] A.H. Christer, Developments in delay time analysis for
modelling plant maintenance, Journal of Operational Research Society 50 (11) (1999) 1120}1137.
85
[2] J.B. Chilcott, A.H. Christer, Modelling of condition-based
maintenance at the coal-face, International Journal of Production Economics 22 (1991) 1}11.
[3] A.H. Christer, W.M. Waller, Delay time models of industrial maintenance problems, Journal of Operational Research 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.
[5] A.H. Christer, W. Wang, J. Sharp, R.D. Baker, A Stochastic
Modelling Problem of High-Tech Steel Production in: A.H.
Christer, S. Osaki, L.C. Thomas (Eds.), Stochastic Modelling in Innovative Manufacturing, Springer, Berlin, 1997,
pp. 196}214.
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 maintenance 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-
* Corresponding author. Tel.: 44-161-745-5000; fax: 44-161745-5559.
ance models, an approximation is generally made
to aid in estimating the number of failures expected
over a PM or inspection period. This paper examines the consequence of this approximation and
presents the modi"ed models relaxing the approximating assumption.
The delay time concept regards the failure process as a two-stage process 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"cation 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.
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 1 1 - 6
78
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
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 approximated. 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 downtime due to failure over a PM period may not be
assumed su$ciently small compared with the cycle
period T. 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.
(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
time d and the plant continues in operation until
&
halted because of another failure or the PM inspection at time ¹ (see Fig. 1). Let NH(¹) denote the
&
expected number of failures over the calendar
period ¹.
Let q and N (q) denote the expected operating
&
time over (0, ¹) and the expected number of failures
over operating time q. Since defects can only arise
or deteriorate when the plant is in operation, the
number of failures NH(¹) arising over calendar time
&
(0, ¹) is given by
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 every ¹ calendar
time units, requires an expected d time units,
1
and all identi"ed defects are repaired.
(2) Inspection is perfect in that, if a defect is present at the time of inspection, it will be identi"ed.
(3) Defects are assumed to arise within the system
at a rate j(u) at operating time u since the last
PM period.
(4) Failures arising during operating time are
identi"ed and repaired immediately with expected downtime d independent of the defect's
&
delay time.
(5) Defects are assumed to only arise, deteriorate,
and lead to failures whilst the system is operating.
(6) The delay time h of a defect is independent
of its time origin and has pdf f (z) and cdf
F(z).
P
NH(¹)"N (q)"
&
&
q
j(u)F(q!u) du,
(1)
u/0
where
q"¹!d N (q).
(2)
& &
Clearly, this requires ¹'d N (q), which may be
& &
considered as a bound on d , or upon the expected
&
number of failures over (0, ¹), which is bounded by
N (¹). If this condition were not valid, either the
&
period T being considered was inappropriate, or the
problem was of an insurance nature when the consequence 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
time q and calendar time ¹.
Fig. 1.
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
If the objective is to select ¹ to minimize the
overall downtime, the objective function to determine the inspection period ¹ to minimize the
expected downtime per unit time over the PM
inspection period, D(¹), is given by
G
H
d N (q)#d
& &
1 ,
(3)
¹#d
1
where q"¹!d N (q) and N (q)":q j(u)F(q!u) du.
0
& &
&
If d ;¹, we may for the purpose of modelling
&
the expected number of failures adopt the convenient practice and assume d "0, in which case
&
Eq. (3) for D(¹) reduces to
D(¹)"
H
G
d N (¹)#d
& &
1 ,
¹#d
1
D(¹)"
(4)
where
P
T
j(u)F(¹!u) du.
0
Of course, having made this approximation, once
the optimal ¹ is 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 is j(u)"j a constant, Eq. (4)
for D(¹) reduces to the well-known form given by
Christer and Waller [3], namely
N (¹)"
&
j¹b(¹)d #d
&
1,
D(¹)"
¹#d
1
where
PA
B
(5)
T ¹!h
f (h) dh.
(6)
¹
0
b(¹) is the probability that a defect arising at random 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 optimal ¹, the simplest procedure is to
assume a q value, and calculate the associated ¹
period and N (q) using Eq. (3). Repetition over an
&
appropriate mesh of q values will establish
parametrically the appropriate ¹ value or range.
This formulation has, of course, assumed the rate
of defects, j(u), to be a function of operating time
b(¹)"
79
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 problem becomes in part a replacement problem [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 downtime for failure estimation, the delay 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 time q be b(q). In
the case of regular inspection on period ¹, by
de"ning operating time appropriately we have from
Christer and Waller [2] that
GP
b(q)"1!
H
= b
+ (1!b)n(1!F(nq!u)) du ,
q
u/0 n/1
(7)
q
where N (q)"jqb(q) and ¹"q#d N (q).
&
& &
With this revised formulation of N (q) to re#ect
&
non-perfect inspection, the non-zero downtime adjusted downtime per unit time model of Eq. (3) still
applies, and as before is readily calculable over
a mesh q.
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 reliability centred maintenance decision-making modelling. Assume for now that a PM inspection returns
the components to a post inspected standard
condition.
Let g(u) be the pdf of the initial time u after an
inspection. We have that the pdf of time t from the
80
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
last inspection to failure in the absence of further
inspection intervention is r(t) where
P
t
g(u) f (t!u) du.
(9)
0
If inspection is undertaken at age ¹, and repair or
replacement to post inspected state initiated immediately upon failure, we have an age-based replacement process in which the expected downtime
per unit time is given by
r(t)"
H
G
d F(¹)#d (1!F(¹)
&
1
.
(10)
(¹#d )(1!F(¹))#d :Ttr(t) dt
1
& 0
In this case, the calendar time ¹ and the operating time q are 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 downtime d as they arise.
&
In this case, if q is the actual operating time, then
the expected number of failures is given by N (q)
&
where
D(¹)"
P
N (q)"
&
q
(1#N (q!u))r(u) du.
(11)
&
u/0
Since ¹"q#N (q)d , the expected downtime per
&
&
unit time model D(¹) is given by
G
H
N (q)d #d
&
1 ,
(12)
D(¹)" &
¹#d
1
where ¹"q#N (q)d , and N (q) is given by
&
&
&
Eq. (11).
4. Numerical examples
The following examples demonstrate the in#uence 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 and d .
1
&
For prototype PM models, we assume that the
pdf of delay time is an exponential distribution,
f (h)"ae~ah with a"0.05 per hour, giving an average delay time of 20 hours, and the average defect
arrival frequency has been taken as j"0.2 defects
per hour. The mean downtime for a failure and PM
are d "0.8 hours and d "0.3 hours respectively.
&
1
The objective function for the downtime has the
form
D(¹)"
Expected number of failures in a cycle(0, ¹)]d #d
&
1.
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 estimating 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
P
T¹!h
1
ae~ah dh"1#
(e~aT!1). (14)
a¹
¹
0
Therefore, the expected total downtime per unit
time, D(¹) is
b(¹)"
A
B
1
j¹ 1#
(e~aT!1) d #d
&
1
a¹
D(¹)"
.
¹#d
1
(15)
4.2. Non-perfect inspection case
It is shown in Eq. (7), that in the case b)1,
P
T = b
+ (1!b)n~1M1!(1!e~a(nT~y))N dy
¹
0 n/1
b =
"1!
+ M(1!b)n~1e~anT(e~aT!1)N
a¹
n/1
=
b
(!e~aT!1) + M(1!b)n~1e~anTN
"1!
a¹
n/1
b(1!e~aT)
"1!
a¹(1!(1!b)e~aT)
b(¹)"1!
0.5(1!e~aT)
"1!
.
a¹(1!(1!0.5)e~aT)
(16)
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
Here, the probability of a defect present during
an inspection being detected, b, is taken as b"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).
81
revised basic PM model is given by
B
A
1
jq 1# (e~aq!1) d #d
&
1
aq
,
D(¹)"
¹#d
1
(20)
where q"¹!jqb(q)d .
&
4.3. Non-homogeneous defect arrival rate case
4.5. Revised non-perfect inspection case
We consider here the non-homogeneous case of
Eq. (1) where defect arrival frequency at time u after
a perfect inspection is given for demonstration
purpose by
In a similar way, b(q) for the imperfect PM model
case from Eq. (16) is
j(u)"0.2!0.06e~0.2u.
b(q)"1!
(17)
Then, the expected number of failures arising in
(0, ¹), N (¹), is
&
P
P
N (¹)"
&
"
T
j(u)F(¹!y) dy
0
T
(0.2!0.06e~0.2y)M1!e~0.05(T~y)N dy,
0
(18)
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 number of defects arising in the PM interval with actual
operating period q is jq, as in the basic model. We
have for the PM interval of length ¹, the number of
failures over an actual operating time (0, q) from
Eq. (6). Therefore, assuming perfect inspection, we
have for b(q)
P
q q!h
1
ae~ah dh"1# (e~aq!1),
aq
q
0
P
q = b
+ (1!b)n~1M1!(1!e~a(nq~y))N dy
q
0 n/1
b(1!e~aq)
"1!
.
aq(1!(1!b)e~aq)
(21)
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
jqb(q)d #d
&
1.
D(¹)"
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 probability f (h) dh. Therefore, the
expected number of failures arising over actual operating time (0, q) in the non-homogeneous defect
arrival rate case is given by Eq. (1), that is, N (q) is
&
given by
P
N (q)"
&
q
0
(0.2!0.06e~0.2y)M1!e~0.05(q~y)N dy.
(19)
(23)
where q"¹!jq b(q)d . It follows from Eq. (2) that
&
the expected total downtime per unit time for the
Therefore, the expected total downtime for the
revised downtime model is given by Eq. (3) where
N (q) is given by Eq. (23).
&
b(q)"
82
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
Now we shall consider the numerical consequences 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 ,
&
is 0.8 hours, the optimal point based on the
Fig. 2. Expected downtime for basic model and revised model.
Fig. 3. Expected downtime for non-perfect inspection model and revising model.
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
83
Fig. 4. Expected downtime for non-Homogeneous case model and revised model.
Table 1
Optimal downtime results of basic model
Non-re"ned model
Re"ned model
Mean downtime
d "0.3
&
d "0.5
&
d "0.8
&
Expected unit downtime
Optimum inspection period
Expected unit downtime
Optimum inspection period
0.0365
19
0.0357
20
0.0484
13
0.0471
14
0.0622
10
0.0601
11
Mean downtime
d "0.3
&
d "0.5
&
d "0.8
&
Expected unit downtime
Optimum inspection period
Expected unit downtime
Optimum inspection period
0.051
19
0.0489
21
0.0713
11
0.0681
13
0.0946
7
0.0891
9
Table 2
Optimal results for downtime of non-perfect inspection case model
Non-re"ned model
Re"ned model
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 failures. 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, ¹ and operating
time, q, are shown in Table 1 and Fig. 2.
84
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
Table 3
Optimal results for downtime of non-homogeneous defect arrival case model
Non-re"ned model
Re"ned model
Mean downtime
d "0.3
&
d "0.5
&
d "0.8
&
Expected unit downtime
Optimum inspection period
Expected unit downtime
Optimum inspection period
0.0342
19
0.0335
20
0.0447
13
0.0436
14
0.0567
10
0.0552
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 for D(¹) are shown for the imperfect PM
case based on the calendar time ¹ and actual operating time q. It can be seen that a larger change in the
optimum values of D(¹) is evident with the re"nement. The optimal interval of the prototype model
based on calendar time ¹, in the case of imperfect
inspection, is 7 hours and the expected total downtime is 0.0946 hours if the mean failure downtime, 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-timebased 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-homogeneous 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 models 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 of D(¹) for the di!erent downtimes
between calendar time and the re"ned actual operating time model resulting from updating b(¹).
Although the percentage savings in total downtime 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/inspection period. Here the characteristic models have
been extended to re"ne the process of estimating
the expected number of failures over a PM period.
From the revised PM model, it is evident that the
expected downtime of failures over PM intervals
may be overestimated by not allowing for downtime in modelling the expected number of failures.
This result is not surprising, since ignoring
A.H. Christer, C. Lee / Int. J. Production Economics 67 (2000) 77}85
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 quality of inspection decreases.
References
[1] A.H. Christer, Developments in delay time analysis for
modelling plant maintenance, Journal of Operational Research Society 50 (11) (1999) 1120}1137.
85
[2] J.B. Chilcott, A.H. Christer, Modelling of condition-based
maintenance at the coal-face, International Journal of Production Economics 22 (1991) 1}11.
[3] A.H. Christer, W.M. Waller, Delay time models of industrial maintenance problems, Journal of Operational Research 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.
[5] A.H. Christer, W. Wang, J. Sharp, R.D. Baker, A Stochastic
Modelling Problem of High-Tech Steel Production in: A.H.
Christer, S. Osaki, L.C. Thomas (Eds.), Stochastic Modelling in Innovative Manufacturing, Springer, Berlin, 1997,
pp. 196}214.