condition-based inspection scheme for condition-based maintenance
TRANSCRIPT
This article was downloaded by: [The University of Texas at El Paso]On: 19 August 2014, At: 08:29Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Production ResearchPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tprs20
Condition-based inspection scheme for condition-based maintenanceHamid Reza Golmakani aa Industrial Engineering Department , Tafresh University , Tafresh , IranPublished online: 21 Sep 2011.
To cite this article: Hamid Reza Golmakani (2012) Condition-based inspection scheme for condition-based maintenance,International Journal of Production Research, 50:14, 3920-3935, DOI: 10.1080/00207543.2011.611540
To link to this article: http://dx.doi.org/10.1080/00207543.2011.611540
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
International Journal of Production ResearchVol. 50, No. 14, 15 July 2012, 3920–3935
Condition-based inspection scheme for condition-based maintenance
Hamid Reza Golmakani*
Industrial Engineering Department, Tafresh University, Tafresh, Iran
(Received 5 September 2010; final version received 14 July 2011)
In condition-based maintenance (CBM) with periodic inspection, the item is preventively replaced if failurerisk, which is calculated based on the information obtained from inspection, exceeds a pre-determinedthreshold. The determination of optimal replacement threshold is often based on minimisation of long-runaverage maintenance costs per unit time due to preventive and failure replacements. It is assumed thatinspections are performed at equal time intervals and that the corresponding cost is negligible. However,in many practical situations where CBM is implemented, e.g. manufacturing processes, inspections requirelabours, specific test devices, and sometimes suspension of operations. Thus, when inspection cost isconsiderable, it is reasonable to inspect less frequently during the time the item is in healthier states, and, morefrequently as time passes and/or the item degrades, namely, a condition-based inspection scheme. This paperproposes a novel two-phase approach for determination of replacement threshold and a condition-basedinspection scheme for CBM. First, it takes into account failure and preventive replacement costs to determinethe optimal replacement threshold assuming that inspections are performed at equal time intervals with nocost. This assumption is, then, relaxed and its consequences on total average cost are evaluated usinga proposed iterative procedure to obtain a cost-effective condition-based inspection scheme. The proposedapproach can be utilised in many CBM applications. For the sake of simplicity of presentation, the approachis illustrated through a simplified case study already reported by some researchers referenced in the paper.
Keywords: maintenance management; condition-based maintenance; condition-based inspections; propor-tional hazard model
Notations
T Random time to failure.Z(t) Value of the stochastic covariate measured at time t.
D Minimum time between inspections.S State space of the process {Z(t)}.
h(t,Z(t)) Hazard function where the system’s age is t and its state is Z(t).� Shape parameter of the Weibull distribution.� Scale parameter of the Weibull distribution.� Weight of the covariate in hazard function.C Preventive replacement cost per one replacement.
CþK Failure replacement cost per one replacement.Cins Cost due to each inspection.
Pij(k) Probability that in inspection, at time (kþ 1)D, state j is observed given that the state at time kD isi and failure may happen after time (kþ 1)D.
P(k) Transition probability matrix associated to kth inspection.d Control limit for preventive replacement.
d � Optimal control limit for preventive replacement.ti Time that the failure risk first reaches the control limit given that the system’s state is i.ki Minimum positive integer number satisfying ti 5 kiD.
�(d) Expected average cost per unit of time due to preventive and failure replacements.Q(d) Probability that failure replacement occurs.
*Email: [email protected]
ISSN 0020–7543 print/ISSN 1366–588X online
� 2012 Taylor & Francis
http://dx.doi.org/10.1080/00207543.2011.611540
http://www.tandfonline.com
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
Q( j, i) Probability of replacement due to failure given that the age of the system is jD and Z( jD)¼ i.W(d) Expected time until replacement, either preventive or failure.
W( j, i) Expected time until replacement given that the age of the system is jD and Z( jD)¼ i.R( j, i, t) Conditional reliability function until time jDþt given that the age of the system is jD and Z( jD)¼ i.
Il Time of inspection l.Zl State observed at inspection l.eI Vector whose elements represent the sequence of the times previous inspections have been performed.eZ Vector whose elements represent the states observed at previous inspections.
ðeI, eZ; rÞ Inspection scheme, where the previous inspection times and the states observed are represented by eIand eZ, respectively, and succeeding intervals for inspections are rD, (rþ 1)D, (rþ 2)D, . . .
TCl ðeI, eZ; rÞ Expected remaining total cost after inspection l, for the inspection scheme represented by ðeI, eZ; rÞ.ECl ðeI, eZ; rÞ Expected remaining cycle time after inspection l, for the inspection scheme represented by ðeI, eZ; rÞ.CPðeI, eZ; rÞ Expected cost per unit time associated to the inspection scheme represented by ðeI, eZ; rÞ.
G(Il,Zl) Expected remaining total cost, given that the system has been operating up to the time of inspection l.Y(Il,Zl) Expected remaining cycle time, given that the system has been operating up to the time of
inspection l.
1. Introduction
Condition based maintenance (CBM) is a maintenance program that recommends maintenance actions based onthe information collected through condition monitoring. CBM attempts to avoid unnecessary maintenance tasksby taking maintenance actions only when there is evidence of abnormal behaviours of a system (or an item)(Jardine et al. 2006). A CBM program, if properly established and effectively implemented, can significantlyreduce unnecessary downtimes, both scheduled and unscheduled, eliminate unnecessary preventive and correctivemaintenance tasks, extend useful life of a system, and reduce system’s total life-cycle cost (Mobley 2002, Akturk andGurel 2007).
The information collected through condition monitoring can be classified into two classes, namely directinformation and indirect information. Direct information is where the measured parameter directly determinesfailure process; for example the thickness of a brake pad, or the wear in a bearing. Indirect information providesassociated information which is influenced by the component condition, but is not a direct measure of the failureprocess; for example, an oil analysis or a vibration frequency analysis (Christer and Wang 1995, Raheja et al. 2006).
Condition monitoring (inspections) can be carried out continuously, periodically, or non-periodically.By continuous monitoring, a machine is continuously monitored and a warning alarm is triggered wheneversomething wrong is detected. In periodic monitoring, inspections are performed at equal time intervals whereas innon-periodic monitoring the times between inspections are not necessarily equal. In both cases, the informationcollected from inspection determines the required maintenance action. This issue will be discussed in more detail inSection 3.
In periodic or non-periodic condition monitoring, the two challenging issues are the determination of (1) thetimes the inspections must be performed and (2) the condition(s) the maintenance action (minimal or major repair orreplacement) must be initiated. Apart from the performance measure selected and the modelling approach utilised,the related research works can be divided into three groups as follows.
The first group of the reported researches focused on the determination of the optimal inspection times(or intervals) only, e.g. Murty and Naikan (1996), Okumura (1997), Chen and Trivedi (2002), Wang (2003), Kallehand van Noortwijk (2006), Wang and Jia (2007). A common assumption is that the information is direct, namely,the state of wear of the underlying deteriorating system can be identified from condition monitoring and, thus, afteran inspection and based on the degree of deterioration an appropriate maintenance action (or no action) isperformed. The objective is, then, to determine the best time to inspect with respect to a given performance criteriasuch as maximum availability or minimum cost per unit time. In Kalleh and van Noortwijk (2006), for example,a decision model for determining the optimal time between periodic inspections for a system with sequential discretestates is suggested. Two scenarios are considered: a scenario in which failure is immediately detected without theneed to perform an inspection and a scenario in which failure is only detected by inspection. For a special casein which the waiting times in each state have identical exponential distributions, an analytical formulation is alsopresented.
International Journal of Production Research 3921
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
The second group of the research works concerned with the dynamic determination of both the inspection times
and the maintenance/replacement threshold, e.g. Kumar and Westberg (1997), Wang (2000), Grall et al. (2002),
Castanier et al. (2003), Dieulle et al. (2003), Chen and Trivedi (2005), Ghasemi et al. (2007), Wang et al. (2009).
In Castanier et al. (2003), for example, the underlying system is subject to a continuous state gradual deterioration
and is monitored by sequential non-periodic inspections. The system is maintained using different maintenance
operations with different effects on the system state, costs, and durations. A multi-threshold policy is used
to schedule the future inspections and the best maintenance actions, using the on-line monitoring information on
the system deterioration level gained from the current inspection subjected to several performance criteria such as
long-run system availability and long-run expected maintenance cost. In Chen and Trivedi (2005), as another
example, a semi-Markov decision model is used for joint optimisation of inspection rate and maintenance policy.
The inspection rate is considered as input parameter to the model. For each individual inspection rate the model
is solved for the optimal replacement policy.The third group of the research works focused on the determination of the optimal maintenance/replacement
threshold only, e.g. Makis and Jardine (1992), Jardine et al. (1997), Banjevic et al. (2001), Chen and Wu (2007),
Lu et al. (2007). A common assumption is that the information obtained from condition monitoring is indirect.
Inspections are performed either periodically or non-periodically but with a pre-determined schedule. The objective
is, then, to determine a threshold that optimises a given performance criteria. In Makis and Jardine (1992), a control
limit policy for a deteriorating system subject to random failure is proposed. In their approach, it is assumed that the
inspections are performed at fixed and given interval of times. The equipment is replaced whenever it fails.
Also, after each inspection and based on the inspection results, if their proposed cost-related failure rate reaches
or exceeds a pre-determined limit, a preventive replacement is scheduled. Proportional Hazard Model (PHM),
(Cox 1972), is used to estimate the system failure rate and the transition probability matrices. The optimal limit is
determined such that the expected maintenance costs per unit time due to preventive and failure replacements over
a long time horizon is minimised. Details of their proposed approach and some case studies have been reported
in Jardine et al. (1999), Banjevic et al. (2001), Jardine et al. (2001), Banjevic and Jardine (2006). It is notable that the
cost of inspection is not considered in their approach.Clearly, higher frequency of inspections can provide more information about the condition of the equipment
and, thus, maintenance actions are performed more effectively, namely, unnecessary preventive replacements are
avoided and the number of replacement due to failure is reduced. Consequently, the cost associated to failure
and preventive replacements are decreased. But in many real cases, inspections require labours, specific test devices,
and sometimes suspension of the operations. Thus, when inspection and analysing the observations is costly,
performing inspections in short intervals incur large inspection cost. In contrast, if inspections are carried out in
long intervals, the cost of failure replacements may significantly increase. This implies that there is a trade-off
between inspection costs and possible failure costs and, thus, by performing inspections at right times it may be
possible to decrease the total average cost of inspections and replacements.Considering the fact that an inspection scheme may significantly influence total average cost in CBM and that
the control limit policy lacks taking into account the inspection cost, in this paper, derivation of a condition-based
inspection scheme within the control limit policy framework is presented. The proposed approach has two steps.
Using the approach developed in Banjevic et al. (2001), Makis and Jardine (1992), the optimal replacement
threshold is first determined. The primary assumption in determining the optimal replacement threshold is that the
inspections are performed at fixed and constant intervals with no cost. This assumption is, then, relaxed and its
consequences on total average cost are evaluated using our proposed iterative procedure to obtain a condition-based
inspection scheme that decreases the total average cost of replacements and inspections, compared to its
traditional constant interval inspection scheme. One may note that the condition-based inspection scheme, obtained
in step two of the proposed approach, is a cost-effective inspection scheme for the optimal replacement threshold,
obtained in step one. The problem of determining simultaneously both the optimal replacement threshold and
the optimal condition-based inspection scheme is a complex problem and can be considered as a possible future
research.The remaining sections of the paper are organised as follows. Section 2 briefly describes the control limit policy.
In Section 3, condition-based inspection scheme is introduced and it is compared with constant-interval inspection
scheme. In Section 4, the approach to determine the condition-based inspection scheme is presented in detail.
In Section 5, a numerical example is given to better demonstrate the proposed approach. In Section 6, some
conclusions are made and, finally, derivations of some equations are detailed in the Appendix.
3922 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
2. Control limit policy for condition-based maintenance optimisation
Makis and Jardine (1992) generalised the CBM policy by introducing a control limit dependent on cost factors fora deteriorating system subject to random failure. It is assumed that failure rate of the system depends both on its ageand on value of stochastic covariate process observable at discrete time points of inspections. The maintenanceaction taken after each inspection is dependent on the condition and age of the system. The optimal replacementpolicy is to replace the system at failure by a new one or perform a preventive maintenance when the failure risk firstreaches or exceeds a control limit, which is determined to minimise the expected maintenance costs per unit time dueto preventive and failure replacement over a long time horizon.
To briefly review the control limit replacement policy, let us consider a non-negative random variable T whichrepresents random time to failure of an item and Z(t) as the value of a non-decreasing stochastic covariate observedat time t. Inspections are performed at times D, 2D, 3D, . . . where D4 0. Let us assume that the state space of theprocess {Z(t)} is finite and Z(kD)2S¼ {0, 1, . . . ,m} for k¼ 0, 1, 2, 3, . . . and Z(0)¼ 0. The state 0 is the best state(new or as new equipment). The state m is the worst state (but the equipment is still working). The non-decreasingassumption of the covariate implies that the state of the equipment cannot become better by itself. It is also assumedthat the process {Z(t), t� 0} is a continuous time discrete process, right continuous with left limits, and Z(t) forkD� t5 (kþ 1)D can be approximated by Z(kD).
Using Cox’s PHM with a baseline Weibull hazard function and the time dependent stochastic covariates todescribe the failure rate of the system, the hazard function is given by
hðt,ZðtÞÞ ¼ �=�ðt=�Þ��1 expð�ZðtÞÞ ð1Þ
and, then, the parameters of the hazard function, i.e. � (shape parameter), � (scale parameter) and � (weight ofcovariate parameter), are estimated using a procedure based on the method of maximum likelihood. The procedureutilises historical failure and suspension data to estimate the parameters (Banjevic et al. 2001).
The method of maximum likelihood is also used to estimate the transition probability matrices for the covariateprocess, m�m matrices P(k) whose elements Pij(k) represents the probability that in the next inspection, i.e., time(kþ 1)D, state j is observed given that the current state is i and failure will not occur by time (kþ 1)D. The estimatedtransition probability matrixes are, then, used in the procedure that obtains the optimal control limit.
Let d4 0 denote the control limit. Each preventive replacement costs C, while each failure replacement costsCþ k, where C, k4 0. The expected cost per unit time due to preventive and failure replacements associated tocontrol limit d, denoted by ’(d), is, then, given by
’ðd Þ ¼Cþ K�Qðd Þ
Wðd Þð2Þ
where Q(d) represents the probability that failure replacement occurs and W(d) is the expected time untilreplacement, either preventive or at failure, given that the failure risk level is equal to d. The optimal control limit,d �, is that value that minimises ’(d) and is determined using the iteration procedure d(n)¼ ’(d(n�1)), n¼ 1, 2, . . . foran arbitrary d(0)4 0, developed by Makis and Jardine (1992). It is proved that d � is a unique value for whichEquation (3) is satisfied.
’ðd �Þ ¼ d � ð3Þ
Also, to compute Q(d) and W(d) in Equation (2), an algorithm in the form of a recursive computationalprocedure is proposed (Makis and Jardine 1992). To briefly review the algorithm, let us denote ti, i2S asti ¼ infft � 0jK� hðt, iÞ � dg for a given d4 0. Also, let ki, i2S be such integers, that ðki � 1ÞD � ti 5 kiD. Letus denote Q( j, i) as the probability of replacement due to failure and W( j, i) as the expected time until replacementgiven that the current age of the system is jD and the current state is i, i.e., Z( jD)¼ i. Assuming Z(0)¼ 0, it is shownin Makis and Jardine (1992) thatW(d)¼W(0, 0) and Q(d)¼Q(0, 0) and the following backward recursion equationscan be used to calculate them,
Qð j, iÞ ¼
0, j4 ki � 1
1� Rðki � 1, i, ti � ðki � 1ÞDÞ, j ¼ ki � 1
1� Rð j, i,DÞ þ Rð j, i,DÞPmr¼i
Qð jþ 1, rÞPirð j Þ, j5 ki � 1
8>>>><>>>>: ð4Þ
International Journal of Production Research 3923
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
Wð j, iÞ ¼
0, j4 ki � 1R ti�ðki�1ÞD0 Rðki � 1, i, sÞds, j ¼ ki � 1R D0 Rð j, i,DÞ þ Rð j, i,DÞ
Pmr¼i
Qð jþ 1, rÞPirð j Þ, j5 ki � 1
8>>>><>>>>: ð5Þ
where the conditional reliability function, R( j, i, t), is given by
Rð j, i, tÞ ¼ expð� expð�iÞ
Z jDþt
jD�=�ðs=�Þ��1dsÞ, 05 t � D: ð6Þ
It is notable that using Equation (6), one can calculate the conditional reliability for next t units of time where05 t�D, given that at Inspection j the state observed is i. In presenting the proposed approach in the next section,calculation of the conditional reliability and its integration for intervals longer than D, namely, t4D, are alsoneeded. In the Appendix, more details and the required equations are given.
3. Constant-interval versus condition-based inspection scheme
In CBM with constant-interval inspection scheme, inspections are performed at equal time intervals, i.e. at times 0,D, 2D, 3D, . . . . At each inspection, the value of the stochastic covariate(s) that is measured through the inspection,i.e. Z(t), is used to calculate the failure rate, h(t, Z(t)), and if it is known that K� h(t,Z(t))� d�, then a preventivereplacement is carried out, otherwise the system continues to operate until, the next inspection. Also, if at any timefailure occurs, a failure replacement is performed. It should be mentioned that it is assumed that the performed tasksby the system during each interval time of D have an equal impact on the system’s degradation process.
In CBM with condition-based inspection scheme, proposed in this paper, inspections are carried out based ona pre-determined scheme, in which inspection times and possible actions to be taken at those inspections are allincluded in advance. In such an inspection scheme, the time of the next inspection is influenced by (1) the age of thesystem at the current inspection, (2) the state of the system observed in the current inspection, and (3) the sequenceof the states observed through the previous inspections. In this scheme, inspections are performed at longer intervalsduring the time the system is in its early age and/or it is in a healthier state, and, are performed at shorter intervals astime passes and/or the system degrades. Also, the sequence of the observed states may influence the time of the nextinspection; a severe change in system’s states, compared to a moderate change, may suggest the next inspections tobe performed at shorter intervals. It is assumed that the time between two consecutive inspections is an integermultiple of D. In Figure 1(a) and Figure 1(b), constant-interval inspection scheme and condition-based inspectionscheme for a typical system are schematically shown.
As it can be seen in Figure 1(a), inspections are set to be performed every D units of time, i.e. inspections I0, I1, I2,I3, . . . are set for times¼ 0, D, 2D, 3D, . . . , respectively. In Figure 1(b), at time 0 the system’s state is 0, i.e. Z(0)¼ 0.At this time, it is determined that the best time for the next inspection, namely Inspection I1, is 4D. At time 4D,inspection is carried out and the state of the system is determined. If it is known that the state of the system is still 0,i.e. Z(4D)¼ 0, the second inspection, namely Inspection I2, is scheduled for time 7D. Similarly, at time 4D, if it isknown that the state of the system has changed to 1, i.e. Z(4D)¼ 1, the second inspection, namely Inspection I2, isscheduled for time 6D. If the state of the system at time 4D is known to be 2, since it is the worst state of the systemno more inspections are needed and, thus, a preventive replacement, highlighted in the figure as PR, is scheduledfor the earliest time where K� h(t,Z(t)¼ 2)� d �. It is notable that it is assumed that the minimum time betweeninspections is equal to D and, thus, as time advances the time between inspections is decreased until it is limited to D.It should also be mentioned that the same replacement policy is maintained, i.e. if at any time failure occurs,a failure replacement is performed and if, at each inspection, it is known that K� h(t,Z(t))� d �, then a preventivereplacement is carried out, otherwise the system continues to operate until the next inspection.
With the assumption that the goal of CBM is cost reduction and that the failure replacement cost is higher thanthe preventive replacement cost, inspections are essentially performed to obtain information about the system’s stateto identify the best time to carry out preventive replacement in order to decrease the cost associated to possiblefailures. If the inspection cost is low or negligible compared to the replacement costs, clearly, performing as manyinspections as possible is preferred, i.e. a very short and constant-interval inspection scheme is desired. On thecontrary, if inspection is costly, when the system is in its early age or it is in a healthier state, the probability that it
3924 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
fails or degrades is usually low and, thus, performing inspections in short intervals may not yield beneficialinformation compared to the cost of inspections. However, decreasing the number of inspections may also increasethe chance that a preventive replacement is not identified and, thus, large failure costs incurred. This implies thatthere is a trade-off between inspection costs and possible failure costs and, thus, by performing a condition-basedinspection scheme it may be possible to decrease the total average cost of inspections and replacements.
One should note that for a given system the proposed condition-based inspection scheme is constructed offlineand, then, is used as a guide to maintain the system accordingly. In the next section, the proposed approach forobtaining the condition-based inspection scheme that yields lower long-run total average cost per unit timecompared to its corresponding constant-interval scheme is described.
4. Proposed approach
4.1 Overview
The proposed approach for obtaining condition-based inspection scheme is a forward iterative procedure. At eachstep, based on the age of the system, the current state of the system, and the sequence of the states observedthrough the previous inspections, the best time for the next inspection is determined, namely a part of the scheme isobtained. The procedure ends once the entire scheme is built.
The determination of the best time for the next inspection is performed by evaluating long-run total average costper unit time corresponding to possible alternative times for the next inspection. The alternative times for the nextinspection are considered as all integer multiples of D like rD such that �þ rD� [tz(�)] where � is the currentinspection time, [�] is floor function, and tzð�Þ ¼ infft � 0jK� hðt,Zð�ÞÞ � d�g:
In calculating the corresponding cost for each alternative time for the next inspection, it is assumed that aftereach alternative time, successive inspections are carried out using a constant-interval inspection scheme with aninterval length of D. Among the alternative times, the time that yields minimum long-run total average cost per unit
Figure 1. (a) Constant-interval inspection scheme and (b) condition-based inspection scheme for a typical system where it isassumed that the value of the stochastic covariate (system’s state) can be 0, 1, or 2. PR is an abbreviation for preventivereplacement.
International Journal of Production Research 3925
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
time is identified as the best time for the next inspection. Having set the time of the next inspection, the procedurerepeats for successive inspection and for all possible states that may be observed in those inspections. The procedureends if it is determined that, for all possible states, a preventive replacement for a future time is recommendedwithout requiring any more inspection. In Figure 2, alternative times that are considered at the first step of theprocedure are schematically shown.
As it is shown in Figure 2, at time zero the state of the system is assumed equal to 0. The Inspection I1can be alternatively performed at times t¼D, 2D, 3D, . . . , rD and so on, where rD� [t0] and t0 ¼ infft � 0jK�hðt,Zð0ÞÞ � d�g. One may note that t0 is the earliest time that while the system’s state remains at State 0 (thehealthiest state), a preventive replacement is recommended for that time. In other words, t0 is the maximum time bywhich the alternative inspection times must be evaluated. Assuming that successive inspections after Inspection I1will be carried out every D units of time, the long-run total average cost per unit time associated to each alternativescheme is calculated (starting from the one where Inspection I1 is done at time D) and, then, is compared to the costcalculated for the last alternative scheme. If it is known that the calculated costs are increasing, examining morealternatives is not needed and the previous scheme identifies the best time for Inspection I1, and if not, examiningother alternatives continues until the condition is met.
At the second step of the procedure, considering all possible states that may be observed at Inspection I1, thesame calculations are applied. As an example, let us assume that in the first step of the procedure, the best time forInspection I1, for the typical system shown in Figure 2, is determined to be at time 3D. Also let us assume thatpossible states to be observed at Inspection I1 can be 0, 1, or 2. At the second step of the procedure, for each of thesepossible states, the best time for the next inspection, i.e. Inspection I2 is determined using the same considerationsexplained in step one, Figure 3.
4.2 Determination of condition-based inspection scheme
In order to present the calculations required for obtaining a condition-based inspection scheme for a systemunder CBM, let us consider a sequence of n inspections and their corresponding system’s states that may beobserved, i.e. a part of a path in the condition-based inspection scheme that is generated until the current iteration.Let eI denote a vector whose elements represent the sequence of the times that inspections have been set to beperformed and eZ denote a vector whose elements represent the states that may be observed at those inspections,i.e. eI ¼ ½I0, I1, I2, . . . , Il, . . . , In�1, In� and eZ ¼ ½Z0,Z1,Z2, . . . ,Zl, . . . ,Zn�1,Zn�: It is assumed that at time I0¼ 0 thestate of the system is Z0¼ 0. Thus, each alternative inspection scheme can be represented by a triplet ðeI, eZ; rÞ.For instance, ðeI ¼ ½I0�, eZ ¼ ½Z0�; 2Þ represents the inspection scheme shown in Figure 2, the second scheme from thetop, and triplet ðeI ¼ ½I0, I1�, eZ ¼ ½Z0,Z1�; 3Þ represents the inspection scheme shown in Figure 3, the third schemefrom the top.
Figure 2. Alternative times for performing Inspection I1, where it is assumed that the system’s state at time 0 is zero.
3926 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
For a given inspection scheme represented by triplet ðeI, eZ; rÞ, let CPðeI, eZ; rÞ denote the expected total cost per
unit time associated to the inspection scheme ðeI, eZ; rÞ. To determine the time for the next inspection, i.e. Inspection
nþ 1, one may note that if the last state observed be the worst system’s state, i.e., Zn¼m, no further inspection
is needed and preventive replacement is scheduled for time tm ¼ infft � 0jK� hðt,mÞ � d�g. Otherwise, CPðeI, eZ; rÞ
is calculated for all r¼ 1, 2, 3, . . . , starting from r¼ 1, and that yields minimum value r for CPðeI, eZ; rÞ identifies the
best time for the next inspection. It is notable that by calculating CPðeI, eZ; rÞ for different values of r, we, in fact,
evaluate how far postponing of the next inspection may yield cost reduction. When an inspection is postponed,
on the one side, one inspection cost is saved. On the other side, we lose the information (the state of the system
obtained from the inspection) that could be useful to lead us to a (possible) preventive replacement. This, in turn,
may increase the chance of a failure occuring and the cost of failure replacement is incurred. In the process of
calculating CPðeI, eZ; rÞ, if it is known that CPðeI, eZ; rÞ � CPðeI, eZ; rþ 1Þ for a given r, it implies that postponing
inspection decreases the total average cost per unit time and, thus, examining the next r is urged. Otherwise, if
CPðeI, eZ; rÞ5CPðeI, eZ; rþ 1Þ, then r is the best one, meaning that no more postponing is beneficial. Thus, in this
case, there is no need to calculate CPðeI, eZ; rÞ for higher values of r. In other words, r� ¼ inffr 2 f1, 2, 3, . . .gjr �
½ðtzn � InÞ=D�,CPðeI, eZ; rÞ5CPðeI, eZ; rþ 1Þg. Once r� is identified, then Inþ1 ¼ In þ r�D.To compute the expected total cost per unit time associated to the inspection scheme represented by ðeI, eZ; rÞ,
i.e., CPðeI, eZ; rÞ, the expected total cost and the expected cycle time associated to the entire scheme must be
computed. Let TCl ðeI, eZ; rÞ and ECl ðeI, eZ; rÞ denote the remaining expected total cost and the remaining expected
cycle time, at the time of inspection l, 0� l� n, respectively. Clearly,
CPðeI, eZ; rÞ ¼TC0ðeI, eZ; rÞ
EC0ðeI, eZ; rÞ: ð7Þ
TC0ðeI, eZ; rÞ and EC0ðeI, eZ; rÞ are computed recursively. As an example, Figure 4 illustrates an inspection scheme
ðeI, eZ; rÞ, where eI ¼ ½I0 ¼ 0, I1 ¼ 4, I2 ¼ 7� and eZ ¼ ½Z0 ¼ 0,Z1 ¼ 0,Z2 ¼ 0�. To determine the time for the next
inspection, i.e. inspection nþ 1¼ 3, TC0ðeI, eZ; rÞ and EC0ðeI, eZ; rÞ for different values of r must be computed.
To compute, for instance, TC0ðeI, eZ; rÞ, TC2ðeI, eZ; rÞ is first calculated and, then, going backward TC1ðeI, eZ; rÞ and
TC0ðeI, eZ; rÞ are subsequently calculated.It can be easily verified that TCl ðeI, eZ; rÞ for 0� l� n� 1 is calculated using the following recursive equation,
where Cins is the cost of each inspection and n is the last inspection index in eI.TCl ðeI, eZ; rÞ ¼ ð1� RðIl=D,Zl, Ilþ1 � Il ÞÞðCþ Kþ lCinsÞ
þ RðIl=D,Zl, Ilþ1 � Il ÞTClþ1ðeI, eZ; rÞ, 0 � l � n� 1
ð8Þ
Equation (8) states that, given the system has survived until the time of inspection l, with the probability of
RðIl=D,Zl, Ilþ1 � Il Þ it will survive until the time of inspection lþ 1 and, thus, its expected remaining total cost will
Figure 3. Alternative times for performing Inspection I2 where it is assumed that the system’s state at time 0 is zero and that thefirst inspection is performed at I1 and State [Z(I)1]2 {0, 1, 2} is observed.
International Journal of Production Research 3927
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
be TClþ1ðeI, eZ; rÞ. Also, with the probability of 1� RðIl=D,Zl, Ilþ1 � Il Þ it will fail before inspection lþ 1 and, thus,
a failure replacement will be performed and the cost will be ðCþ Kþ lCinsÞ.It can also be shown that TCl ðeI, eZ; rÞ for l¼ n, namely, TCnðeI, eZ; rÞ is computed using Equation (9).
TCnðeI, eZ; rÞ ¼ 1� RlnD,Zn, rD
� �� �ðCþ Kþ nCinsÞ þ R
lnD,Zn, rD
� �
�
Pm�1i¼ZnðPrÞZn,i
ðCþ KÞQðIn=Dþ r, iÞ
þC 1�QðIn=Dþ r, iÞð Þþ
ðnþ 1ÞCins þWðIn=Dþ r, iÞCins=D
0B@1CA
þ ðPrÞZn,m
II1ðIn þ rD,mÞðCþ ðnþ 1ÞCinsÞ þ ð1� II1ðIn þ rD,mÞÞ
�RðIn=Dþ r,m, tm � ðIn þ rDÞÞðCþ ðnþ 1ÞCinsÞ
þ ð1� RðIn=Dþ r,m, tm � ðIn þ rDÞÞÞðCþ Kþ ðnþ 1ÞCinsÞ
!0B@
0BBBBBBBBB@
1CCCCCCCCCAð9Þ
In Equation (9), II1(t,Z(t)) is an indicator function which is 1 if K� h(t,Z(t))� d � and 0 otherwise. Also, ðPrÞZn,i
is the probability that, at time Inþ rD, the state of the system changes to i given that the state at time In is Zn.Similarly, it can be easily verified that ECl ðeI, eZ; rÞ for 0� l� n� 1 is calculated using the following recursive
equation.
ECl ðeI, eZ; rÞ ¼
Z Ilþ1�Il
0
RðIl=D,Zl, tÞdtþ RðIl=Zl, Ilþ1 � Il Þ
� EClþ1ðeI, eZ; rÞ
ð10Þ
Where
ECnðeI, eZ; rÞ ¼
Z rD
0
RðIn=D,Zn, tÞdt
þ RðIn=D,Zn, rDÞ
Xm�1i¼Zn
ðPrÞZn,iWðIn=Dþ r, iÞ
þ ðPrÞZnmð1� II1ðI1 þ rD,mÞÞ
R tm�ðInþrDÞ0 RðIn=Dþ r,m, tÞdt
0BBB@1CCCA: ð11Þ
Figure 4. Illustration of expected remaining total cost and expected remaining cycle time for a typical inspection scheme ðeI, eZ; rÞ,where eI ¼ ½0, 4, 7� and eZ ¼ ½0, 0, 0�.
3928 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
It is notable that in Equations (9) and (11), ðPrÞzn,i is the probability that, after rD units of time, the state of thesystem changes to i given that the system has survived until the current time, that the current state is Zn, and that thefailure will not happen by the next rD units of time. It should be mentioned here that, for simplicity of presenting therequired equations, the same conditional transition probability matrix, P, is assumed for all intervals of the length D.Thus, for the interval of the length rD, where r¼ 1, 2, 3, . . . , the conditional transition probability matrix is simplyequal to Pr. This assumption does not limit the scope of the work because its relaxation only entails the replacementof the matrix P with, let’s say, P(k) for each interval kD to (kþ 1)D, k¼ 0, 1, 2 . . . .
To elaborate Equations (8) to (11), let us refer to the inspection scheme ðeI, eZ; rÞ shown in Figure 4, whereeI ¼ ½0, 4, 7� and eZ ¼ ½0, 0, 0�. If this inspection scheme, namely, inspecting the system at times0, 4D, 7D, 7Dþ rD, 7Dþ ðrþ 1ÞD, . . . is used for maintaining the system, depending on when the system fails orwhen it is preventively replaced, the total cost and the cycle length vary. However, their expected values, namely,TC0ðeI, eZ; rÞ and EC0ðeI, eZ; rÞ, are of interest. With the probability of 1� RðIl=D,Zl, Ilþ1 � Il Þ, the system failsbefore the time of the first inspection, i.e., 4D. Alternatively, it may survive by that time with the probabilityof RðIl=D,Zl, Ilþ1 � Il Þ, where l¼ 0. In the former case, the total cost incurred is (CþKþ lCins) and in thelatter case, it is TC1ðeI, eZ; rÞ. This is what Equation (8) states. Similarly, for the former case the expected cycle
time isRIlþ1�Il
0
RðIl=D,Zl, tÞdt and for the latter case, it is EC1ðeI, eZ; rÞ, Equation (10).
Given that the system has survived by the time of I1¼ 4D and that the state observed at this inspection is Z1¼ 0with the probability of 1� RðIl=D,Zl, Ilþ1 � Il Þ, it fails before the time of the second inspection, i.e. 7D.Alternatively, it may survive by that time with the probability of RðIl=D,Zl, Ilþ1 � Il Þ, where l¼ 1. In the formercase, the total cost incurred is (CþKþ lCins) and in the latter case, it is TC2ðeI, eZ; rÞ, Equation (8). Similarly, for theformer case the expected cycle time is
R Ilþ1�Il0 RðIl=D,Zl, tÞdt and for the latter case, it is EC2ðeI, eZ; rÞ, Equation (10).
Given that the system has survived by the time of I2¼ 7D and that the state observed at this inspection is Z2¼ 0,with the probability of 1� R I2
D ,Z2, rD� �
, it fails before the time of the next inspection, i.e. 7Dþ rD. Alternatively, itmay survive by that time with the probability of R I2
D ,Z2, rD� �
. In the former case, the total cost incurred is(CþKþ nCins), where n¼ 2, first part of Equation (9). In the latter case, the total cost depends on the state thatis observed at the time of 7Dþ rD. ðPrÞzn,i represents the probability that the system’s state changes from Zn toi ¼ Zn,Zn þ 1, . . . ,m. Two situations are examined.
First: if the state changes to m, i.e., the worst state. If I2 þ rD � tm, i.e., indicator function II1ðI2 þ rD,mÞ ¼ 1,since a preventive replacement is immediately carried out at that time, the total cost incurred is (Cþ (nþ 1) Cins).If II1ðI2 þ rD,mÞ ¼ 0, a preventive replacement is scheduled for time tm and no more inspection is needed. Thus,with the probability of RðI2=Dþ r,m, tm � ðI2 þ rDÞÞ, the system survives by the time tm and the correspondingtotal cost will be (Cþ (nþ 1) Cins). Also, it fails with the probability of 1� RðI2=Dþ r,m, tm � ðI2 þ rDÞÞ and thecorresponding total cost will be (CþK (nþ 1) Cins), last part of Equation (9).
Second: if the state changes to a state other than m referring to Equations (4) and (5), QðIn=Dþ r, iÞ andWðIn=Dþ r, iÞ are the probability that the cycle ends with a failure and the expected remaining cycle length,respectively, given that the inspections are performed every D units of time from and after the inspection In=Dþ r.Thus, the inspection cost is ðnþ 1ÞCins þWðIn=Dþ r, iÞCins=D . If the cycle ends with failure, the cost is (CþK),otherwise; it is C. Therefore, the expected total cost is ðCþ KÞQðIn=Dþ r, iÞ þ Cð1�QðIn=Dþ r, iÞÞ þðnþ 1ÞCins þWðIn=Dþ r, iÞCins=D, middle part of Equation (9).
4.3 Calculation of expected total cost per unit time for a given condition-based inspection scheme
In the previous sub-section, it was shown how a condition-based inspection scheme for a system under CBM can beobtained. Based on the steps the proposed approach generates the condition-based inspection scheme, the expectedtotal cost per unit time associated to the obtained inspection scheme is clearly less than the case where inspectionsare performed at equal time intervals. In this sub-section, the equations required for calculating the expected totalcost per unit time for a given condition-based inspection scheme is presented. By knowing that, one can compute thepercentage of cost reduction the condition-based inspection scheme can achieve.
Let us consider a given condition-based inspection scheme, for example the scheme shown in Figure 1(b). Eachpath in the scheme ends with a preventive replacement. The system may go through any of the paths, be replacedpreventively, or fail at any time. Thus, depending on the path taken, the time a failure occurs, or the time apreventive replacement is performed, the cost incurred and the length of the time the system has been operatingvaries. In order to calculate the expected total cost per unit time, let G(Il, Zl) and Y(Il, Zl), for l� 0, denote the
International Journal of Production Research 3929
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
expected remaining total cost and the expected time to replacement, respectively, given that the system has been
operating up to the time of Inspection l, i.e., time Il and given that the state observed at this time is ZðIl Þ ¼ Zl in the
given path. Denoting the expected total cost per unit time by ETC, it is clear that
ETC ¼ Gð0, 0Þ=Yð0, 0Þ: ð12Þ
Let M denote the maximum number of inspections performed among the paths. Let us define the indicatorfunction II2ðIl Þ which is 1 if Inspection l is the last inspection in a given path and 0 otherwise. The following
backward recursive equations, Equations (13) and (14), are used to calculate G(Il, Zl) and Y(Il, Zl) for
l 2 f0, 1, . . . ,Mg:
GðIl,Zl Þ ¼ II1ðIl,Zl ÞðCþ lCinsÞ þ ð1� II1ðIl,Zl ÞÞ
�
II2ðIl Þð1� RðIl=D,Zl, tzl � Il ÞÞðCþ Kþ lCinsÞ
þRðIl=D,Zl, tzl � Il ÞðCþ lCinsÞ
� �þ ð1� II2ðIl ÞÞ
�1� RðIl=D,Zl, Ilþ1 � Il ÞÞðCþ Kþ lCinsÞ
þRðIl=D,Zl, Ilþ1 � Il ÞPm
i¼zlðPðIlþ1�Il Þ=DÞzl,iGðIlþ1,iÞ
!0BBBB@
1CCCCA, ð13Þ
YðIlZl Þ ¼ ð1� II1ð1� II1ðIlZl ÞÞ
�
II2ðIl ÞR tzl�Il0 RðIl=D,Zl, tÞdtþ ð1� II2ðIl ÞÞ
�
R Ilþ1�Il0 RðIl=D,Zl, tÞdtþ RðIl=D,Zl, Ilþ1 � Il Þ
�Pm
i¼zlðPðIlþ1�Il Þ=DÞzl,iYðIlþ1, iÞ
!0BB@1CCA ð14Þ
5. A numerical example
Let us consider transmission systems of large transporters in a mining company. The estimation of hazard rate
function and transition probability matrices from historical data for this system have been reported in Banjevic and
Jardine (2006) as a case study. However, for the clarity of presenting the proposed approach, we have reduced the
number of system’s states and used different hypothetical failure and suspension data and, then, following the same
approach, we have estimated the hazard function’s parameters, i.e. beta, eta, and gamma as well as the transition
probability matrix. At each inspection, oil analysis is carried out and the value of iron in oil (as the most significant
covariate) is measured. Having made discrete the iron values into covariate bands, the state of the transmission isidentified as 0, 1, or 2. The process {Z(t). t� 0} has, thus, three states, i.e. S¼ {0, 1, 2}. The minimum time between
two inspections is one month, i.e. D¼ 1. The hazard function for the transmission system has been estimated as
hðt,ZðtÞÞ ¼ �=�ðt=nÞ��1 expð�ZðtÞÞ: where � ¼ 2:323, � ¼ 21:457, and � ¼ 0:827. Let us also assume that a preventive
replacement costs 10 k, a failure replacement costs 30 k, and each inspection costs 2 k, i.e. C¼ 10, K¼ 20, and
Cins¼ 2. Also, it is assumed that the transition probability matrix for all time intervals of the length D is equal and
has been estimated as
P ¼
0:749 0:251 0:000
0:000 0:811 0:189
0:000 0:000 1:000
0BB@1CCA:
Employing Equations (2) to (6), the optimal control limit is first obtained: d �¼ 1.8635 Table 1 shows the
calculations at each step of the algorithm.Using the proposed approach, the complete condition-based inspection scheme is now constructed. We start
at time 0 assuming that the state is 0, i.e.eI ¼ ½0� and eZ ¼ ½0�. In order to determine the time for the first inspection,
we calculate CPðeI, eZ; rÞ, starting from for r¼ 1. Here, t0¼ 19.1561 and D¼ 1. Thus, the maximum r for which
CPðeI, eZ; rÞ may be calculated is [t0/D]þ 1. The first r for which CPðeI, eZ; rþ 1Þ4CPðeI, eZ; rÞ determines r�,
i.e., r � ¼ inffr � 1jCPðeI, eZ; rþ 1Þ4CPðeI, eZ; rÞg, , Equations (7) to (11). Table 2 shows these calculations. As it can
3930 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
be seen from Table 2, CPðeI, eZ; 10þ 1Þ4CPðeI, eZ; 10Þ. Thus, r�¼ 10 and, i.e., the best time for the first inspectionis I1 ¼ 10.
The states that may be observed at the first inspection, which is set to be performed at time I1 ¼ 10, can beZ1 ¼ 0, Z1 ¼ 1 or Z1 ¼ 2. The failure risk corresponding to these states, at that time, areK� hð10, 0Þ ¼ 0:788585 d �, K� hð10, 1Þ ¼ 1:803045 d �, and K� hð10, 2Þ ¼ 4:12256 � d �, respectively. Thus,the system is preventively replaced if Z1¼ 2 is observed at Inspection I1 and otherwise, i.e. if States 0 or 1 isobserved, it is kept on operation. However, since t1 � I1 ¼ 10:2525� 105D, if Z1¼ 1, a preventive replacement isscheduled for time 10.2525 and no more inspection is needed in this path.
At the second iteration, using Equations (7) to (11) again (where eI ¼ ½0, I1 ¼ 10�and eZ ¼ ½0,Z1 ¼ 0�) CPðeI, eZ; rÞfor different values of r is calculated, Table 3.
As it can be seen from Table 3, CPðeI, eZ; 4þ 1Þ4CPðeI, eZ; 4Þ and, thus, the best time for the second inspection isI2 ¼ I1 þ 4 ¼ 14. The states that may be observed at Inspection I2 ¼ 14 can be Z2 ¼ 0, Z2 ¼ 1, or Z2 ¼ 2.Clearly, if Z2 ¼ 2, a preventive replacement is performed. The failure risk computed for States 0 and 1 areK� hð14, 0Þ ¼ 1:230765 d � and K� hð14, 1Þ ¼ 2:81406 � d �, respectively. Thus, at this inspection if State 1 isobserved, a preventive replacement is also performed. If State 0 is observed, the system continues operation.
In the next iteration, eI ¼ ½0, I1 ¼ 10, I2 ¼ 14� and eZ ¼ ½0,Z1 ¼ 0,Z2 ¼ 0�. Again, Equations (7) to (11) are usedto determine the time for the third inspection, Table 4.
From the calculation given in Table 4, CPðeI, eZ; 3þ 1Þ4CPðeI, eZ; 3Þ and, thus, r� is 3 and the best time for thenext inspection is I3¼ 17. The states that may be observed at Inspection I3¼17 can be Z3 ¼ 0, Z3 ¼ 1, or Z3 ¼ 2.Clearly, if Z3 ¼ 1, or Z3 ¼ 2, a preventive replacement is performed. If Z3 ¼ 0, the failure risk isK� hð17, 0Þ ¼ 1:591215 d� and, thus, the system continues operation.
Table 2. Calculations to determine the time for the first inspection assuming eI ¼ ½0� and eZ ¼ ½0�.r TC0ðeI, eZ; rÞ EC0ðeI, eZ; rÞ CP0ðeI, eZ; rÞ r TC0ðeI, eZ; rÞ EC0ðeI, eZ; rÞ CPðeI, eZ; rÞ
1 29.2756 7.5777 3.8634 7 19.0992 7.9141 2.41332 26.9650 7.5765 3.5590 8 18.9587 8.2446 2.29953 24.7009 7.5707 3.2627 9 19.2321 8.6020 2.23584 22.6617 7.5566 2.9989 10 19.8590 8.9684 2.21435 20.9210 7.5303 2.7782 11 21.1754 9.4330 2.24486 19.6797 7.6273 2.5802
Table 1. Calculations to determine optimal control limit.
d t0 t1 t2 k0 k1 k2 W(0,0) Q(0,0) ’(d)
5 40.3918 21.6181 11.5703 41 22 12 10.2107 0.6240 2.20162.2016 21.7286 11.6294 6.2242 22 12 7 8.0149 0.2504 1.87261.8726 19.2267 10.2903 5.5075 20 11 6 7.5902 0.2072 1.86361.8636 19.1563 10.2526 5.4873 20 11 6 7.5777 0.2061 1.86351.8635 19.1561 10.2526 5.4873 20 11 6 7.5777 0.2061 1.8635
Table 3. Calculations to determine the time for the secondinspection assuming eI ¼ ½0, 10� and eZ ¼ ½0, 0�.r TC0ðeI, eZ; rÞ EC0ðeI, eZ; rÞ CPðeI, eZ; rÞ
1 25.1956 10.7274 2.34872 24.5468 10.8485 2.26273 24.3940 11.0460 2.20844 24.6530 11.2849 2.1846
5 25.2179 11.5371 2.1858
Table 4. Calculations to determine the time for the thirdinspection assuming eI ¼ ½0, 10, 14� and eZ ¼ ½0, 0, 0�.r TC0ðeI, eZ; rÞ EC0ðeI, eZ; rÞ CPðeI, eZ; rÞ
1 27.3862 12.0848 2.26622 26.9943 12.1692 2.21823 27.0321 12.3054 2.1968
4 27.3957 12.4677 2.1973
International Journal of Production Research 3931
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
To find the best time for the fourth inspection, Equations (7) to (11) are again used with eI ¼ ½0, 10, 14, 17� andeZ ¼ ½0, 0, 0, 0�, for r¼ 1, 2, 3, Table 5.As it can be seen in Table 5, CPðeI, eZ; 2þ 1Þ4CPðeI, eZ; 2Þ. Thus, I4¼ 19. At this inspection, if Z4¼ 1 or Z4¼ 2
a preventive replacement is performed. If Z4¼ 0, since t0¼ 19.15165 20, examining higher value of r is not neededand a preventive replacement is scheduled for time 19.1516 and the procedure stops. The completed inspectionscheme is shown in Figure 5.
Using Equations (12) to (14), the expected total cost per unit time associated to the condition-based inspectionscheme shown in Figure 5 is calculated, G(0,0)¼ 19.7381, Y(0,0)¼ 9.0037, and thus, ETC¼ 2.1922. One may notethat the expected total cost per unit time for constant-interval inspection scheme (with intervals of the length equalto D) is equal to d �D þ
Cins
D ¼ 3:8635. In other words, performing the condition-based inspection scheme obtainedfrom the proposed approach, compared to the constant-interval inspection scheme, can reduce the total cost per unittime by 43.26% for the case in the example.
In order to examine how changes in the values of Cins and K may affect the obtained inspection scheme,the procedure explained above is repeated with the same data but assuming values of 1, 2, 3, 4, 5, and 9 for Cins andvalues of 20, 30, 40, 50, and 100 for K, Table 6. It should be noted that if inspection cost is assumed to be 0,the proposed approach generates a same constant-interval inspection scheme and, thus, it is not included in thetable.
As it is shown in Table 6, for each value of K, t0, t1, t2 and d � is first determined. Then, for each Cins, thecondition based inspection scheme is obtained. Subsequently, the expected cost per unit time for both the constant-interval inspection scheme (CIIS) and the condition-based inspection scheme (CBIS) is calculated and finally thepercentage of cost reduction (PCR) is computed. One should note that the value of C is equal to 10 in all the casesin the table.
By considering the values of PCR in each column, it can be seen that as the inspection cost increases, thepercentage of cost reduction also increases, meaning that the approach generates a more cost-effective inspectionscheme if the ratio Cins
C increases. Similarly, by considering the values of PCR in each row, it can be concluded that asthe ratio Cins
K increases the percentage of cost reduction also increases. In all the cases, the percentage of costreduction is positive. By considering the obtained inspection schemes, one may note that as the inspection costincreases the number of inspections in the obtained inspection scheme decreases and/or the time betweenconsecutive inspections increases. In fact, as expected, the inspection scheme is formed such that the total averagecost of inspection and replacements is decreased.
Figure 5. Condition-based inspection scheme for the system in the example. PR is an abbreviation for preventive replacement.
Table 5. Calculations to determine the time for the fourth inspectionassuming eI ¼ ½0, 10, 14, 17� and eZ ¼ ½0, 0, 0, 0�.r TCðeI, eZ; rÞ ECðeI, eZ; rÞ CPðeI, eZ; rÞ
1 28.1957 12.6260 2.23312 27.9738 12.6937 2.2038
3 28.5775 12.9182 2.2122
3932 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
Table
6.ComparisonbetweenConstant-IntervalInspectionSchem
e(C
IIS)andCondition-BasedInspectionSchem
e(C
BIS),assumingdifferentvalues
forK
andCins.
PCR
isanabbreviationforPercentageofCost
Reduction.
K¼20,
t 0¼19.1561,t 1¼10.2526,
t 2¼5.4873,d�¼1.8635
K¼30,
t 0¼15.4477,t 1¼8.2678,
t 2¼4.4250,d�¼2.1028
K¼40,
t 0¼13.2900,t 1¼7.1129,
t 2¼3.8069,d�¼2.2977
K¼50,
t 0¼11.8414,t 1¼6.3376,
t 2¼3.3920,d�¼2.4655
K¼100,
t 0¼8.3264,t 1¼4.4564,
t 2¼2.3851,d�¼3.0944
Schem
e[0913161819]
[0710121415]
[07101213]
[06911]
[0468]
Cins¼1
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
2.8635
2.0740
27.57%
3.1028
2.3357
24.72%
3.2977
2.6215
20.51%
3.4655
2.7810
19.75%
4.0944
3.4604
15.48%
Schem
e[010141719]
[081215]
[071013]
[06911]
[057]
Cins¼2
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
3.8635
2.1922
43.26%
4.1028
2.5045
38.96%
4.2977
2.7582
35.82%
4.4655
2.9463
34.02%
5.0944
3.7355
26.67%
Schem
e[0101519]
[0913]
[071113]
[0711]
[058]
Cins¼3
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
4.8635
2.2615
53.50%
5.1028
2.6532
48.01%
5.2977
2.8934
45.38%
5.4655
3.1608
42.17%
6.0944
3.9813
34.67%
Schem
e[01117]
[0914]
[0812]
[0711]
[058]
Cins¼4
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
5.8635
2.3475
59.96%
6.1028
2.7421
55.07%
6.2977
3.0495
51.58%
6.4655
3.2898
49.12%
7.0944
4.1845
41.02%
Schem
e[01118]
[0914]
[0813]
[0711]
[068]
Cins¼5
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
6.8635
2.4025
65.00%
7.1028
2.8264
60.21%
7.2977
3.1646
56.64%
7.4655
3.4187
54.21%
8.0944
4.4588
44.92%
Schem
e[0
14]
[011]
[010]
[09]
[07]
Cins¼9
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
CIIS
CBIS
PCR
10.8635
2.5814
76.24%
11.1028
3.1411
71.71%
11.2977
3.5843
68.27%
11.4655
3.9210
65.80%
12.0944
5.1413
57.49%
International Journal of Production Research 3933
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
6. Summary and conclusions
With the assumption that the goal of CBM is cost reduction and that failure replacement cost is usually higher
than preventive replacement cost, inspections are essentially performed to obtain information about the system’sstate to identify the best time to carry out preventive replacements to decrease the cost associated to possible
failures. If inspection cost is low or negligible compared to replacement costs, clearly, performing as manyinspections as possible is preferred. However, when cost of inspection and analysing information is considerable, on
the one hand performing inspections in short intervals incur large inspection cost. On the other hand, decreasing thenumber of inspections may also increase the chance that a preventive replacement is not identified and, thus, large
failure cost is incurred. This implies that there is a trade-off between inspection costs and possible failure costs.This paper aims to emphasise the significance of an inspection scheme for CBM and to show how selecting an
appropriate condition-based inspection scheme may reduce the cost associated to a CBM program. The paperproposes an approach in which preventive and failure replacement costs as well as inspection costs are taken into
account to determine the replacement threshold and a cost-effective condition-based inspection scheme. The controllimit replacement policy, already reported by some researches and referenced in this paper, is first utilised to
determine the optimal replacement threshold. Having relaxed the assumption that inspections are performed atequal time intervals with no cost, an iterative procedure is, then, used to evaluate alternative inspection schemes and
their associated total average costs to obtain the condition-based inspection scheme. One may note that this paperassumes that the performed tasks by the system during equal time intervals have an equal impact on the system’s
degradation process. As a potential future research, however, one may relax this assumption, since there are caseswhere utilising the system with higher rate would result in greater deterioration of the system and, thus, more
frequent inspections would be beneficial.
References
Akturk, M.S. and Gurel, S., 2007. Machining conditions-based preventive maintenance. International Journal of Production
Research, 45 (8), 1725–1743.Banjevic, D. and Jardine, A.K.S., 2006. Calculation of reliability and remaining useful life for a Markov failure time process.
IMA Journal of Management Mathematics, 17 (2), 115–130.Banjevic, D., et al., 2001. A control-limit policy and software for condition-based maintenance optimisation. INFOR, 39 (1),
32–49.
Castanier, B., Berenguer, C., and Grall, A., 2003. A sequential condition-based repair/replacement policy with non-periodic
inspections for a system subject to continuous wear. Applied Stochastic Models in Business and Industry, 19 (4), 327–347.
Chen, D. and Trivedi, K.S., 2002. Closed-form analytical results for condition-based maintenance. Reliability Engineering and
System Safety, 76 (1), 43–51.Chen, D. and Trivedi, K.S., 2005. Optimisation for condition-based maintenance with semi-Markov decision process. Reliability
Engineering and System Safety, 90 (1), 25–29.Chen, A. and Wu, G.S., 2007. Real-time health prognosis and dynamic preventive maintenance policy for equipment under aging
Markovian deterioration. International Journal of Production Research, 45 (15), 3351–3379.Christer, A.H. and Wang, W., 1995. A simple condition monitoring model for a direct monitoring process. European Journal of
Operational Research, 82 (2), 258–269.
Cox, D.R., 1972. Regression models and life-tables. Journal of the Royal Statistical Society, Series B (Methodological), 34 (2),
187–220.Dieulle, L., et al., 2003. Sequential condition-based maintenance scheduling for a deteriorating system. European Journal of
Operational Research, 150 (2), 451–461.Ghasemi, A., Yacout, S., and Ouali, M.S., 2007. Optimal condition based maintenance with imperfect information and the
proportional hazards model. International Journal of Production Research, 45 (4), 989–1012.Grall, A., Berenguer, C., and Dieulle, L., 2002. A condition-based maintenance policy for stochastically deteriorating systems.
Reliability Engineering and System Safety, 76 (2), 167–180.
Jardine, A.K.S., Banjevic, D., and Makis, V., 1997. Optimal replacement policy and the structure of software for condition-based
maintenance. Journal of Quality in Maintenance Engineering, 3 (2), 109–119.
Jardine, A.K.S., et al., 2001. Optimising a mine haul truck wheel motor’s condition monitoring program: Use of proportional
hazards modeling. Journal of Quality in Maintenance Engineering, 7 (4), 286–301.Jardine, A.K.S., Joseph, T., and Banjevic, D., 1999. Optimising condition-based maintenance decision for equipment subject to
vibration monitoring. Journal of Quality in Maintenance Engineering, 5 (3), 192–202.
3934 H.R. Golmakani
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4
Jardine, A.K.S., Lin, D., and Banjevic, D., 2006. A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mechanical Systems and Signal Processing, 20 (7), 1483–1510.
Kalleh, M.J. and van Noortwijk, J.M., 2006. Optimal periodic inspection of a deterioration process with sequential conditionstates. International Journal of Pressure Vessels and Piping, 83 (4), 249–255.
Kumar, D. and Westberg, U., 1997. Maintenance scheduling under age replacement policy using proportional hazards modeland TTT-plotting. European Journal of Operational Research, 99 (3), 507–515.
Lu, S., Tu, Y.C., and Lu, H., 2007. Predictive condition-based maintenance for continuously deteriorating systems. Quality andReliability Engineering International, 23 (1), 71–81.
Makis, V. and Jardine, A.K.S., 1992. Optimal replacement in the proportional hazards model. INFOR, 30 (1), 172–183.
Mobley, R.K., 2002. An introduction to predictive maintenance. New York: Butterworth- Heinemann.Murty, A.S.R. and Naikan, V.N.A., 1996. Condition monitoring strategy – A risk based interval selection. International Journal
of Production Research, 34 (1), 285–296.
Okumura, S., 1997. An inspection policy for deteriorating processes using delay-time concept. International Transactions inOperational Research, 4 (5–6), 365–375.
Raheja, D., et al., 2006. Data fusion/data mining-based architecture for condition-based maintenance. International Journal ofProduction Research, 44 (14), 2869–2887.
Wang, W., 2000. A model to determine the optimal critical level and the monitoring intervals in condition based maintenance.International Journal of Production Research, 38 (6), 1425–1436.
Wang, W., 2003. Modelling condition monitoring Intervals: A hybrid of simulation and analytical approaches. Journal of the
Operational Research Society, 54 (3), 273–282.Wang, L., Chu, J., and Mao, W., 2009. A condition-based replacement and spare provisioning policy for deteriorating systems
with uncertain deterioration to failure. European Journal of Operational Research, 194 (1), 184–205.
Wang, W. and Jia, X., 2007. An empirical Bayesian based approach to delay time inspection model parameters estimation usingboth subjective and objective data. Quality and Reliability Engineering International, 23 (1), 95–105.
Appendix: Calculation of R( j, i, t) andR t0Rð j, i, sÞds
The conditional reliability, R( j, i, t), indicates the probability of survival until time jDþ t, (05 t�D), knowing that the failurehas not happened until time jD, and the state of the equipment at the time of jD is i. It is given by (Cox 1972),
Rð j, i, tÞ ¼ exp �
Z jDþt
jDhðs,ZðsÞÞds
� �, 05 t � D ðA1Þ
where h(s,Z(s)) is hazard function. As in Banjevic et al. (2001), we use the hazard function hðs,ZðSÞÞ ¼ expð�ZðsÞÞ ��
� �s�
� ���1where ZðsÞ 2 0, 1, . . . ,mf g. Thus, Equation (A1) can be re-written as
Rð j, i, tÞ ¼ exp �
Z jDþt
jDexpð�ZðsÞ
�
�
� �s
�
� ���1ds
!, 05 t � D: ðA2Þ
Equation (A2) is not valid for t4D, because the state of the system may change from i to any states i, iþ 1, . . . , m at the endof each successive interval D. The transition probability matrixes for the covariate process, namely, m � mmatrixes P(k), k¼ 0, 1,2, . . . are estimated using historical data (Banjevic et al. 2001). Each element Pij(k) represents the probability that in nextinspection, i.e. time (kþ 1)D, state j is observed given that at the moment of inspection k, i.e., time kD, the state is i and that thefailure will not occur by the time (kþ 1)D. For the sake of simplicity in presenting the proposed approach, a same transitionprobability matrix, P, for all intervals of D is used in this paper. This assumption does not limit the scope of the work because itsrelaxation only entails the replacement of the matrix P with P(k) for intervals kD to (kþ 1)D, k¼ 0, 1, 2, . . . .
To calculate the conditional reliability for t4D, one should note that if the equipment survives for a period of t4D after jD,it has to have survived until ( jþ 1)D, which may happen with the probability of R( j, i, D). At this time, the state of the equipmenttransfers from i to any states i, iþ 1, . . . , m, with a probability of Pi,i,Pi,iþ1,Pi,iþ2, . . .Pi,m, respectively. Thus, for t4D, we canconclude that Rð j, i, tÞ ¼ Rð j, i,DÞ
Pml¼i ðPÞilRð jþ 1, l, ðt� DÞÞ. Therefore, the conditional reliability at time jD, while the state is
Z( jD)¼ i, is given by
Rð j, i, tÞ ¼
exp �R jDþtjD expð�iÞ ��
� �s�
� ���1ds
� �, 05 t � D
Rð j, i,DÞPml¼i
ðPÞilR jþ 1, l, ðt� DÞð Þ, t4D
8>><>>: ðA3Þ
Since calculation of R( j, i, t) for t4D has to be done recursively, its integration, i.e.R t0 Rð j, i, sÞds, can be easily calculated
numerically, namely, computing R( j, i, s) (using Equation A.3) for very small intervals and summing them up.
International Journal of Production Research 3935
Dow
nloa
ded
by [
The
Uni
vers
ity o
f T
exas
at E
l Pas
o] a
t 08:
29 1
9 A
ugus
t 201
4