First International Conference on Emerging Trends in Engineering and Technology
Machinery Condition Monitoring System Selection –
A Multi-Objective Decision Approach using GA
A.K. Verma*, A. Srividya*, P.G. Ramesh@
* Indian Institute of Technology, Bombay @ Defence Institute of Advanced Technology, Pune
[email protected]
maintenance effects[6-8]. Soft computing techniques
and Genetic algorithms are increasingly being used in
maintenance decision problems as they provide an
efficient platform for both modeling and solution
phases of the problem[9-13]. Most models in literature
use deterioration levels of machinery systems to
schedule inspection or condition monitoring. But, in
many cases, particularly for large engineering plants, it
is not possible to precisely identify the level of
deterioration since there are several components in the
plant and each of them has several modes of
deterioration. Further, there are interdependencies
between the components. Therefore, for a maintenance
decision maker on site, it is preferable to identify the
state of the sub-system by means of the level of repairs
that are necessary to restore the prevailing state to a
“as-good-as-new” state. There is also a need to
optimize the characteristics of the condition monitoring
system at a macro level. This will enable decisions
regarding the area and extent of effective applicability
of CBM for a large plant consisting of several subsystems arranged in a complex way.
Abstract
In this paper, a framework for Condition Based
Maintenance (CBM) of large engineering systems has
been suggested to address the need to optimize the
characteristics of both the condition monitoring system
and the plant that is being maintained at a macro level.
CBM for large engineering systems has been modeled
using Markov process. The issues which are critical to
CBM, namely, predictability, detectability as well as
implications of availability and cost have been
considered in the framework to provide effective
decision support. The application of the framework has
been demonstrated using a numerical example. The
trade offs between the Pareto optimal solutions of
various objectives have been studied and the effects of
variation of the maintenance objective function values
with detectability and predictability have been brought
out.
.
1. Introduction
Maintenance has an essential role in ensuring that
engineering systems perform at expected levels of
reliability, availability and safety in a cost effective
manner and in keeping with the corporate demands. In
the case of Maintenance of Large Engineering Plants
(LEP) like, aircraft, shipboard machinery, power plants
and nuclear installations, owing to their peculiarities,
Time Based Maintenance (TBM) is preferred as an
effective strategy [1]. But TBM is based on the
assumption that failures are age related which is
contrary to a study where it is shown that about 80% to
85% of equipment failures are caused due to the effects
of random events that happen in the system [2].
Therefore it is necessary to augment TBM of large
engineering plants with Condition Based Maintenance
(CBM) so as to optimally realize the objectives of
maintenance. Features of condition monitoring,
diagnostic and prognostic systems have been discussed
in literature[3-5]. There has been a widespread use of
Markov and semi-Markov processes for studying
978-0-7695-3267-7/08 $25.00 © 2008 IEEE
DOI 10.1109/ICETET.2008.127
This paper presents a framework for providing
decision support to select condition monitoring
systems based on their detectability and predictability
and provide for them at appropriate locations of the
system. In section 2 the CBM model is described as a
discrete space, continuous time Markov process. In
sections 3 and 4, objective and constraint functions are
developed. This is followed by formulation of
expressions for detectability and predictability of the
condition monitoring system in section 5. A numerical
example is presented in section 6. Results and
Conclusions are discussed in sections 7 and 8
respectively.
2. Model description
Consider a Large Engineering Plant (LEP) in
which n modules are subjected to Condition Based
Maintenance (CBM). Each of the n modules of the
plant is subjected to a particular condition monitoring
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arrangement. The condition monitoring arrangements
comprise of one or more techniques, including manual
inspection and assessment of condition of the module.
Each of the n modules is essential for the plant to be in
the operational state.
In respect of each of the n modules, initially,
periodic monitoring of its condition is done at a
particular monitoring rate. When an exceeding of the
threshold values of one or more of those parameters
indicating the healthy condition of the module
(Relevant Condition Indicators, RCI [16]) is detected
by the condition monitoring system, the module is
subjected to a more detailed condition monitoring
strategy. In this strategy, prediction of time to carry out
preventive maintenance so as to prevent a catastrophic
failure is done using the monitored parameters
(Relevant Condition Predictor, RCP[16]).
Again,
when crossing of threshold value of the RCP of the
module is detected by the condition monitoring system,
the module is subjected to an appropriate level of
preventive maintenance (PM).
If the RCP or RCI is detected but the instant of
failure is not detected by the monitoring system
sufficiently in advance, the PM of the module is
delayed for necessary logistic preparation. On the other
hand, if the exceeding of an RCI or an RCP is not
detected,
the module
fails
catastrophically,
necessitating Corrective Maintenance (CM). Both PM
and CM restore the module to “as good as new”
condition.
The model provides for the ability to enable
regression to a previous operationally better state in
case the measured deterioration value reduces due to
state changes or cessation of some external disturbing
factors [14]. The model is shown schematically in
Figure 1. The above CBM problem is modeled as a
discrete space, continuous time Markov process as
shown in Figure 2. In constructing the model and
formulating the objective functions, the discussions in
reference [15] have been taken forward to include
logistic delay and a detailed analysis of detectability
and predictability features of condition monitoring
system on the maintenance objectives of LEP.
Pi,p
Di,d
Fi
λAi,aBi,π
LEP – n modules under CBM
Y
Is condition
within
threshold
value ?
N
Increased rate/or
continuous/monitoring/
condition prediction
Stopped – Level 1 prev.
maintenance
Y
Is condition
within
threshold
value ?
N
Required
maintenance
level
Log
delay
Stopped – Level 2 prev.
maintenance
Catastrophic Failure –
Corrective Maintenance
Log
delay
Figure 1. Schematic diagram showing CBM of LEP
Maintenance (CBM) of the plant, and the proportion of
probabilities of CM to PM.
A
A 11,1,1
A
A 11,a, a
A
A
1
1,A
,A
F
F1
1
B
B1 ,1
1 ,1
B
B1,
1 ,π
π
D
D 1 ,1
1 ,1
3. Notation
The condition of the module in each state of the
Markov model is determined by the extent or level of
repairs or maintenance required to restore it to as good
as new condition. It is required to minimize the
Unavailability, overall cost of Condition Based
P
P 1 ,1
1 ,1
B
B 1 ,W
1 ,W
D
D 1 ,d
1,d
D
D 1 ,D
P
P 1 ,p
P
P 1 ,P
1 ,p
1 ,D
1 ,P
Figure 2. Markov model of CBM of a LEP
Further, if δi,π ∈ [0,1] is the detectability feature
of the relevant condition monitoring arrangement π of
module i, and αi ,π and βi ,π are the failure rates of the
module i while undergoing condition monitoring of
Table 1. Description of states of the Markov model
i
Ai,a
1,2,….,A
Module i under condition monitoring strategy
‘π’, П = 1,2,….,W
Module i under preventive maintenance of level
p , p = 1,2,….,P
Module i held up for preventive maintenance
due to logistic delay of category d, d = 1,2,….,D
Module i in the catastrophic failure state
Transition rate from state Ai,a to Bi,π and so on
for all other state transitions.
Bi,π
Number of the module, i = 1,2,……,n
Operational state ‘a’ of module, ‘i' , a =
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strategy π, then the transition rates to the states of
higher levels of condition monitoring and various
categories of logistic delays are respectively,
λAi,aBi,π = δi,π αi ,π
(1)
λAi,aDi,d = Li,l δi,π βi ,π
(2)
Where, Li,l is the probability of the module i
undergoing level ‘l’ of PM. Also, the corresponding
transition rates to the states of catastrophic failure are
λAi,aFi,π = (1-δi,π) αi ,π
(3)
(4)
λAi,aFi,π = (1-δi,π) βi ,π
The transition from the states of logistic delays
will depend on the predictability. Higher the
predictability, faster will be the rate of transition from
the state of logistic delay to that of PM. If ρ ∈ [0,1] is
the predictability feature of the condition monitoring
system of an impending failure, then,
(5)
λDi,dPi,p = Li,l ρ δi,π βi ,π
4.
Formulation
constraint functions
4.1
of
objective
preventively when the condition demands or when the
component of module fails. This can be given by


 P

F
1
1
(10)
T = min  ∑
,∑
; i = 1, 2...., n 
Rp
(f )
P
F
(
)(
)
 p =1
λ

=
1
f
i
F
(r )
i
 P ( Pi , p )(∑ λPi , p )

r =1


where λP( r ) is the rth transition rate out of state Pi,p and
i,p
λF( f ) is the fth transition out of state Fi (f = 1,2,…,F).
i
C Plant =
P
W
C π δ π + Cρ
∑
π
i,
i,
p =1
,i
ρi + Ci P ( Fi ) +
=1
D
∑C
i , p P ( Pi , p ) + ∑ Ci , d P ( Di ,d )
d =1
}
(11)
Where Ci ,π , Cρ ,i , Ci , Ci , p and Ci ,d are the cost related
to the detectability of the condition monitoring
system, predictability of the condition monitoring
system, cost of catastrophic failure, cost of PM and the
cost due to logistic delay respectively.
and
Objective functions
4.1.3
The
ratio,
(Prob(CM)/Prob(PM))Plant
The ratio of the sum of the probabilities of
CM and the sum of probabilities of PM in respect of all
the modules of the plant is proposed an indicator
towards the efficacy of the condition monitoring
system.
4.1.1. Unavailability. The
steady
state
unavailability of the plant is given in respect of the
probabilities of catastrophic failures of the constituent
modules and the down time of the plant due to logistic
delay. Since all modules are necessary for the
operation of the plant, the steady state unavailability of
the module is given by:
D


(6)
U Module =  P( Fi ) + ∑ P ( Di , d ) 
d =1


And, for the entire plant,
n
D
 

U Plant = 1 − ∏ 1 −  P( Fi ) + ∑ P( Di ,d )  


d =1
i =1 
1 n
∑{
T i =1
n
 CM 

=
 PM 
∑ P( F )
(12)
i
n
i =1
P
∑∑ P( P
i, p
)
i =1 p =1
4.2 Constraint Function
(7)
The CBM model for the plant has to integrate with
the Time Based Maintenance or Scheduled Preventive
Maintenance (SPM) Model. One of the approaches
proposed for the same is by trying to ensure that the
Mean Time Between Failures (MTBF) derived from
the CBM model is greater or equal to the SPM interval,
tSPM . The least MTBF of all the modules must be
greater than or equal to the SPM interval and this is
proposed as a constraint to the optimization problem.
4.1.2 Cost.
The cost of CBM is dependent on the
initial and operational cost of the condition monitoring
system as represented in this model by their
detectability and predictability. Higher detectability
and predictability imply higher cost. Cost is incurred
when the plant is down for CM owing to not only the
lost operational time, but also the secondary damages
suffered and the safety compromised. Cost is also
incurred due to the logistic delay, again due to the lost
operational time as well as the extra effort and
resources required for emergency arrangements for
repairs and maintenance. A short logistic delay is
possible if adequate warning of failure is given by the
condition monitoring system, which in turn depends on
its predictability. Further, the cost of CBM is proposed
as a time rate which will be the rate of cost over an
operational window of duration T. As per the CBM
policy, replacement or repair or maintenance is done


 A

1
min ∑
; i = 1, 2,..., n  ≥ t SPM
Ra
 a =1

(r)
 P ( Ai ,a )(∑ λ Ai ,a )



r =1
(13)
4.3 Constrained Multi-Objective Optimisation
Problem
The CBM problem modeled above is solved as a
Constrained Multi-Objective Optimisation Problem.
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(b) Diagnostic Ability
Let the set of parametric and non-parametric
symptoms gathered by the condition predicting system
(17)
be Pi ,π = Pi ,π ,1 , Pi ,π ,2 ,...., Pi ,π , j ,...Pi ,π , J
The optimization problem is solved using Genetic
Algorithms (GA). GA is proposed since it is suitable
for handling non linear multiple objectives with several
variables and requiring to be optimized simultaneously
with each objective maintaining its identity. On
execution of the GA, a population of solutions is
obtained each of which is optimum with trade offs
[10]. Each optimal solution corresponds to a set of
decision parameters, namely, in the present case,
detectability and predictability factors in respect of
each module. The decision maker can choose the
optimum solution that is acceptable for the given
multi-criteria situation. Domain information pertaining
to the failure rates, repair rates, inspection intervals,
relative probabilities various levels of PM are
incorporated in the problem to narrow down the
decision space and aid in the search for the optima.
{
Si ,π = { All possible combinations of symptoms that
are found to occur together to yield
meaningful
diagnosis}
f : Si ,π → Di ,
Define
(19)
Si ,π ⊆ Pi ,π
σ ie,π + σ ia,π = 1
(22)
where,
(23)
σ
d
i ,π
+σ
p
i ,π
=1
A plant consists of two modules, both of which are
required for the functioning of the plant. The CBM
decision model proposed above is applied to the plant,
appropriately adopting the Markov diagram in Figure
2. The values of the constants used in the problem are
αi ,π = 0.033; βi ,π = 0.05; λAi,aBi,π = 1 & 0.166; λBi,πAi,a = 1
& 4; λFi,Ai,a = 0.06; λPi,pAi,a = 1. For simplicity, the
failure rates are assumed to be identical for both the
modules. The constrained multi-objective optimization
problem was solved using Non-Dominated Sorting
Genetic Algorithm (NSGA-II) [18]. The parameters for
the GA are based on experimental/conservative values.
The steady state Probabilities and other terms in
the objective and constraint functions are obtained by
solving the matrix equation, 24 [19].
these are essential for the predicting system to provide
maximum lead time for preparation for maintenance
and thereby reducing logistic delays and enabling
better planning of maintenance. Following quantitative
measures are proposed for determining Predictability:
(a) Prognostic ability:
ρip,π = P(ξi −Ti,π ) <<ε × P(Ti,π −tid,π ) ≥ LDT;Ti,π < tSPM (16)
where, ε is a small positive number t
Predictability: ρi ,π = σ id,π ρid,π + σ ip,π ρip,π
6. Numerical Example
time.
Predictability is proposed to be further divided into
prognostic ability ρip,π and diagnose-ability. Both of
7. Results
}
d
i ,π
(21)
Further, it is proposed that both the detectability and
predictability be given as the weighted sum of their
constituent parts.
e
a
(22)
Detectability : δ i ,π = σ i ,π eπ + σ i ,π aπ
where µi = E[ξi ] , ξi is the failure time random variable
of module i and Ti ,π is the preventive replacement
{
(20)
Ai ,π ,a ∈Si ,π
a =1,2,....α
(15)
µi
Ai ,π ,a ∈ Si,π
a = 1, 2,....α
Thus, diagnostic ability is given by
ρid,π = ∑ P (Ai ,π , a ) P ( f ( Ai ,π , a ) givenAi ,π , a )
The overall outputs of the condition monitoring
arrangements can be divided into detectability
(representing RCI) and predictability (RCP) factors.
Detectability is further divided into effectiveness and
accuracy. Effectiveness eπ is the proportion of the
expected number of failures avoidable by technique π
relative to the total expected number of failures under
Failure Replacement Policy (FRP). Accuracy aπ of the
same strategy is defined as the proportion of total
expected number of failures under Failure
Replacement Policy to the expected number of total
removals. For the long run they are defined as [17] :
µ P(ξi ≤ Ti,π )
(14)
eπ = 1 − i
E[min(ξi , Ti,π )]
E[min(ξ i , Ti ,π )]
Ai ,π ,a → Di ,k
Also,
5. Detectability and predictability
aπ =
}
The set of diagnosis leading to preventive maintenance
(18)
be Di = { Di ,1 , Di ,2 ,..., Di , k ,...Di , K }
The result of multi-objective optimisation using
the GA is a set of optimal solutions which form Pareto
optimal fronts in the objective spaces. Each solution
corresponds to a set of values for the objectives, which
in turn correspond to a set of decision variables. All
is the time of
detection of the impending fault or failure of module i
using the technique π and LDT is the logistic delay
time.
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-λ −λ
( 1,1)  A1,1B1,1 A1,1B1,1
PA
PB

λA1,1B1,1
 ( 1,1) 
PA
( 1,2) 0


( 1,2) =0
PB

PP
( 1,1)  0

 
( 1,1) 
PD
0
PF

 ( 1)  1

λB A
0
λP A
0
λFA 
−λB1,1A1,1 −λB1,1A1,2 0
0
0
0
0
1,1 1,1
1 1,1
λB A
−λA1,2B1,2 −λA1,2F1 λB1,2A1,2
0
λA B
0
0
0
−λP1,1A1,1 λD1,1P1,1 0
0
0
λB D
0
−λD1,1P1,1 0
1
1
1
1
1
1,1 1,2
1,2 1,2
0
0
0
−λB1,2A1,2 −λB1,2D1,1 0
0
0
1,2 1,1
decision maker. Likewise, Figure 4 shows the variation
of cost with unavailability. Figure 5 gives the relation
between the ratio of probabilities CM/PM and
unavailability. It can be seen from Figure 6 that the
variation of average predictability is more rapid than
that of average of detectability with respect to
unavailability. In other words, availability is relatively
more sensitive to predictability. Extending this
observation, detectability is more important in the early
stages of monitoring which will improve reliability and
safety of the plant and predictability is more important
−1
0
1,1 1,1
1











0
0

0

0
0

0
1

(24)
solutions are optimal, one differing from the other due
to the trade off between the objectives. The trade offs
between the relative cost rate of CBM and the ratio of
probabilities of CM/PM is shown in Figure 3. It can be
350
300
350
250
Prob(CM)/Prob(PM)
300
P rob (C M )/P rob (P M )
250
200
200
150
100
150
50
100
0
0.1
50
0.15
0.2
0.25
0.3
0.35
Unavailbaility
0
55
65
75
85
95
105
115
Figure 5. Unavailability of the plant vs. the ratio of
the probabilities of CM and PM
Relative Cost Rate
Figure 3. The trade offs between the relative cost
rate of CBM and the ratio of probabilities of
CM/PM
1.05
Average P red ictability and Detectability
0.95
115
105
Relative Cost Rate
95
85
0.85
0.75
0.65
0.55
0.45
0.35
75
0.25
65
0.1
0.15
0.2
0.25
0.3
Unavailability
Ave Predictability
55
0.1
0.15
0.2
0.25
Ave Detectability
Figure 6. Averages of detectability and
predictability of both modules compared against
Unavailability
0.3
0.35
Unavailability
Figure 4. Trade off between Unavailability of the
plant and relative cost rate of CBM
in the later stages, which will improve steady state
availability.
Finally, the decision maker can select the most
acceptable solution from among the Pareto optimal
solutions using the decision statements for
seen that the cost of CBM increases along with greater
demand for PM in preference to CM. This result is on
expected lines. However, the results give more precise
decision support in respect of the two objectives to the
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0.35
predictability and detectability (equations 14-23),
which in turn can be ascertained based on statistical
data or expert opinion or both.
[7]
J.S. Ivy and H.B. Nembhard, “A Modeling
Approach to Maintenance Decisions Using Statistical Quality
Control and Optimization”, Quality and Reliability
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8. Conclusions
[8]
D. Chen and K.S. Trivedi, “Optimisation for
Condition Based Maintenance with Semi Markov Decision
Process”, Reliability Engineering and System Safety, 2005 pp
177-187.
CBM is aimed at preventing catastrophic failures
of equipment and providing sufficient lead time for
planning, preparing and undertaking preventive
maintenance. Selection of a condition monitoring
system would require decisions regarding the adequacy
of detectability and predictability which would provide
the necessary reliability and availability of the plant at
affordable cost. In this paper, a framework for the
above purpose has been provided by modeling CBM as
a Markov process where detectability and
predictability features of the condition monitoring
system are incorporated as decision variables. The
objectives of minimizing unreliability, cost and the
extent of corrective maintenance compared to
preventive maintenance have been simultaneously
optimized using GA. The trade offs between the
objectives as well as sensitivity of detectability and
predictability to the objectives were studied. The
decision maker can select that optimum solution in the
light of the corresponding detectability and
predictability.
[9]
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P.G. Busacca., “Multi Objective Optimisation by
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177-187.
792
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