-
A Multi-Objective Genetic Algorithm for determining
efcientRisk-Based Inspection programs
Mrcio das Chagas Moura a,n, Isis Didier Lins a, Enrique Lpez
Droguett b,Rodrigo Ferreira Soares c, Rodrigo Pascual d
a CEERMA Center for Risk Analysis, Reliability and Environmental
Modeling, Federal University of Pernambuco, Recife, PE, Brazilb
Center for Risk and Reliability, Mechanical Engineering Department,
University of Maryland, College Park, USAc PETROBRAS S.A., Brazild
Physical Asset Management Lab, Department of Mining Engineering,
Ponticia Universidad Catlica de Chile, Santiago, Chile
a r t i c l e i n f o
Article history:Received 31 January 2014Received in revised
form17 August 2014Accepted 15 September 2014Available online 28
September 2014
Keywords:Inspection programsRisk reductionRisk-Based
InspectionMulti-Objective Genetic Algorithm
a b s t r a c t
This paper proposes a coupling between Risk-Based Inspection
(RBI) methodology and Multi-ObjectiveGenetic Algorithm (MOGA) for
dening efcient inspection programs in terms of inspection costs
andrisk level, which also comply with restrictions imposed by
international standards and/or localgovernment regulations. The
proposed RBIMOGA approach has the following advantages: (i) a
user-dened risk target is not required; (ii) it is not necessary to
estimate the consequences of failures;(iii) the inspection
expenditures become more manageable, which allows assessing the
impact ofprevention investments on the risk level; (iv) the
proposed framework directly provides, as part of thesolution, the
information on how the inspection budget should be efciently spent.
Then, geneticoperators are tailored for solving this problem given
the huge size of the search space. The ability of theproposed
RBIMOGA in providing efcient solutions is evaluated by means of two
examples, one ofthem involving an oil and gas separator vessel
subject to internal and external corrosion that causethinning. The
obtained results indicate that the proposed genetic operators
signicantly reduce thesearch space to be explored and RBIMOGA is a
valuable method to support decisions concerning themechanical
integrity of plant equipment.
& 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Past accidents in several types of industries have
demonstratedthat equipment malfunction is one of the major causes
of unex-pected and undesirable events such as toxic and
inammabledischarges, re and explosions. Failures to function
properly areusually due to inadequate integrity management systems
thatmay result in cracks, holes, ruptures, and consequently loss
ofcontainment of dangerous substances. Therefore, integrity
controlhas been used for guaranteeing aging machineries work in
anappropriate manner, assuring plant safety against adverse
occur-rences [1].
In this context, inspection has been used as a technique
toexamine the real situation of equipment exposed to
damagemechanisms (e.g., thinning, stress corrosion cracking,
high-temperature hydrogen attack, mechanical fatigue, brittle
fracture),thus reducing the uncertainty of its condition. The aim
is toidentify these potential damage mechanisms and steer efforts
in
order to prevent failures by prioritizing systems that need
moreattention.
Decisions about which equipment should be investigated,which
inspection approach will be performed, and when thisevent will take
place have become intricate problems due to thecomplexity of the
involved processes, especially in reneries andpetrochemical
industries. Thus, Risk-Based Inspection (RBI) hasbeen used to
support the decision makers in managing theschedule of those
interventions. The fundamental principle ofRBI is quite simple:
after a user-dened risk target Rt (i.e. theacceptable risk) has
been chosen, the inspection program isdetermined in order to not
allow the risk level to exceed Rt ,thereby avoiding loss of
containment and, subsequently, unwel-come effects [2].
Thus, RBI deals with the consequences of holes and ruptures
inpressurized equipment in terms of the area affected by the
out-come of the possible release of dangerous materials, and then
theexpenses to execute mitigation solutions to the problems
causedby these occurrences. Then, a baseline curve for the risk
levelR k Pf k UFC is estimated over time k by combining the
prob-ability of failure Pf k and the respective nancial consequence
FCfor each equipment under analysis.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ress
Reliability Engineering and System Safety
http://dx.doi.org/10.1016/j.ress.2014.09.0180951-8320/& 2014
Elsevier Ltd. All rights reserved.
n Corresponding author. Tel./fax: 55 81 2126 7112.E-mail
address: [email protected] (M.d.C. Moura).
Reliability Engineering and System Safety 133 (2015) 253265
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In this context, Singh and Markeset [3] proposed an RBIplanning
based on fuzzy logic approach for oil and gas carbonsteel pipelines
subject to CO2 corrosion. Khan et al. [4] used the RBImethodology
to develop inspection and maintenance strategiesthat maximize
system availability; the approach is applied to thesteam generating
system of an oil eld thermal power plant. Chienet al. [5] developed
a semi-quantitative RBI analysis for pressuresafety valves (PSV)
used in lubricant process units. Li et al. [6]developed an RBI
theoretical framework for ship structures using adecision tree
method. In Tien et al. [7], an RBI-based model forpiping systems
has been built in accordance with internationalstandards and local
government regulations; the purpose was toprovide
inspection-related personnel with the optimal planningtools, to
enable effective predictions of potential piping risks, andthen to
enhance the degree of safety in petrochemical industries.Marangone
and Freire [8] applied the RBI methodology in themanagement of
mechanical integrity of an oil and gas separatorvessel subject to
corrosion mechanisms. Vinod et al. [9] appliedthe RBI methodology
for an H2S-based process plant along with anapproach devised for
handling the inuence factor related to thequantity of H2S released.
Shuai et al. [10] used the RBI technologyto assess quantitatively
the risk of crude oil tanks in an oil depotin China.
RBI quantitative analysis yields a tool that enables the
recognitionof the actual equipment's situation, and hence the
intervention needsto reduce the risk exposure. Then, the estimation
of the risk level R k is updated from the data gathered at each
inspection epoch in order tocontinue representing the current
condition of the system. In this way,the recommended practice [11]
provides guidance on developing anRBI program for xed equipment
(including pressure vessel, piping,tankage, pressure relief
devices, and heat exchanger tube bundles) inrening, petrochemical,
and chemical process plants. Thus, Ref. [11]aims at providing
quantitative calculation methods to determine aninspection plan.
Nevertheless, all the aforementioned works have fourmain
disadvantages:
(i) The user-dened threshold Rt for risk level is not taken
intoaccount as an objective to be optimized. In other words,
therisk measure should be maintained below Rt , but
nothingguarantees that this parameter is chosen in the most
efcientway. On the contrary, the decision maker solves this
problemthrough a trial-and-error method by changing the
inspectionprogram until the risk target is no longer reached.
(ii) This process of not allowing the risk level R k to go above
thepre-set target Rt also disregards the costs associated with
theassigned inspection program. For instance, the risk limit Rt
mayrequire an unreasonably high budget if it is chosen to be too
low.
(iii) The calculation of the risk measure R k Pf k UFC at
sometime k involves the determination of Pf k along with
FCconsidering the loss of containment of a particular
equipment.Even though the screening of critical equipment is
usuallycarried out using a simplied qualitative approach, Vinodet
al. [9] pointed out that, in general, the number of compo-nents to
be considered for a quantitative RBI assessment isstill very large.
Furthermore, the estimation of FC is a verylaborious task because
of the amount of information required.In fact, determining FC
involves the estimation of costs ofequipment repair and
replacement, costs of damage to sur-rounding equipment in affected
areas, costs associated withproduction losses and business
interruption because of down-time to repair or replace damaged
equipment, costs due topotential injuries associated with a failure
and environmentalcleanup costs. These categories of costs are added
up in orderto estimate FC.
(iv) Finally, it is not possible to point out how and when
theinspection resources should be spent. This means that no
guideline on how to plan the inspections is provided, i.e.,
howto determine which techniques have to be adopted and whenthey
have to be performed in full compliance with interna-tional
standards and/or local government regulations. Forexample, decision
makers may face problems on how tocombine inspection techniques: is
it better to perform anumber of low-cost/low-effective inspections
or fewerinspections with higher effectiveness, but more
expensive?By effectiveness of an inspection technique, we mean
theability of detecting and measuring damage mechanisms.
Therefore, this paper proposes an original RBI
multi-objective-based framework, which aims at minimizing both the
total risklevel and the costs related to the inspection program.
Hence, atrade-off analysis is required since both objectives are
conicting.In other words, given a planning horizon, the idea is to
nd theoptimal compromise between risk and inspection costs, and
thusovercome the aforementioned drawbacks as follows:
(i) The risk target Rt is no longer required a priori because
risk isnow an objective to be minimized. In this way, the
decisionmaker will not be concerned about the inspection policy
thatmaintains the risk level below Rt . In fact, the
inspectionprogram will be one of the results of the proposed
approach.Moreover, even if the risk must be below a preset target
valueRt (possibly to comply with safety regulations), the
proposedmulti-objective problem also allows taking into account a
riskconstraint.
(ii) The inspection costs are also handled as an objective.
There-fore, as both risk and expenditures with inspection
activitiesare now considered, the resulting inspection program will
beof great signicance on balancing safety, availability
andinspection cost requirements.
(iii) Despite the large number of equipment that should
beprioritized in a rening or petrochemical facility as well asthe
information requirements to estimate the nancial con-sequences,
note that FC is considered constant for a particularequipment
according to Ref. [11]. In this way, only Pf k variesover time k
and is updated based on data collected frominspection, implying
that Pf k and R k curves have the sameshape. Thus, in our
multi-objective approach, the demandingstep of computing FC is no
longer necessary because we willdirectly work on Pf k instead of R
k , which is equivalent;even if a risk constraint has to be
satised, as commented initem (i), it can be considered a posteriori
as FC becomesavailable;
(iv) Finally, the proposed framework provides the
decision-makerwith the information on which inspection technique
shouldor should not be performed at each time-step, which in
turndenes the inspection program. This piece of information willbe
associated with each pair of solution (risk or probability
offailure; inspection costs), directly identifying which
actionsshould be followed for managing risk to a minimal level.
Thus, in the context of RBI, the multi-objective approachemerges
as an alternative to handle the conicting objectives ofrisk (or
probability of failure) and inspection costs so as to createefcient
inspection policies that comply with regulation standards.In a
multi-objective optimization, a solution that optimizes
allobjectives concurrently is very difcult to be reached or it
simplydoes not exist. In this way, instead of having a unique
solution asin single objective cases, one may obtain a set with
multiplesolutions. These solutions, named non-dominated, present
acompromise among objectives and usually do not yield an
optimalvalue for either of them individually. Once this set is
obtained,
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265254
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decision makers can choose any of its elements based on
theirpreferences, and then implement the selected solution.
Depending on the number of periods considered within
theinspection planning horizon and on the quantity of
inspectiontechniques available, the number of possible inspection
plans canbe prohibitively large for an exhaustive evaluation of
their perfor-mance. Therefore, a probabilistic approach will be
developed forthe quest of inspection programs representing the
optimal com-promise between both objectives. Heuristic optimization
methods,such as Genetic Algorithms (GAs), have interesting
characteristicsto handle multi-objective optimization problems [12]
because(i) they are population-based, i.e., many potential
solutions aresimultaneously considered; (ii) they permit a
separated treatmentof different objectives, thus not requiring any
transformation ofthe multiple objectives into a unique function.
GAs [13] attempt toimitate computationally the natural evolution
process in which thettest individuals are more likely to remain in
the population. Inthe optimization context, an individual is a
potential solution ofthe considered problem and a set of
individuals is the population,which evolves according to some
genetic-based operators, such asselection, crossover and
mutation.
Then, in this work, a novel Multi-Objective Genetic
Algorithm(MOGA) is developed to minimize both likelihood of failure
andcost subject to constraints imposed by international
standardsand/or local regulations. The generation of the initial
population,crossover and mutation are operators devised to create
onlyfeasible inspection plans. In this way, the search space to
beexplored by MOGA is reduced, unnecessary evaluations of
theobjectives are circumvented and penalty functions are
notrequired to handle unfeasible solutions.
The remainder of the paper is organized as follows. Section
2summarizes the RBI methodology. The multi-objective problem
ischaracterized and formulated in Section 3. Section 4 presents
theMOGA along with the genetic operators tailored for solving
theproblem and an overview of the coupling RBIMOGA. In Section
5,the ability of the proposed RBIMOGA in providing accurate
solutionsis rst evaluated for a hypothetical example. Then, in
Section 6, theproposed RBIMOGA approach is applied to an oil and
gas separatorvessel subject to internal and external corrosion
causing thinning.Finally, Section 7 provides some concluding
remarks.
2. Risk-Based Inspection
Usually in an RBI study, a qualitative assessment is
primarilyperformed in order to obtain a categorized risk level for
the consideredequipment. Probabilities and consequences of failures
are assigned topre-set categories, and then combined into a risk
matrix. In general,equipment with medium high and high risk levels
are submitted to amore detailed quantitative risk analysis [14].
Then, inspectionresources should be allocated by prioritizing these
items.
Therefore, the objective of this section is to present the
back-ground of the quantitative RBI methodology that draws from
Ref.[11]. More specically, it is summarized how the likelihood
offailure Pf k and nancial consequences FC are estimated in orderto
provide the risk level R k over time k.
2.1. Probability of failure
The probability of failure for a given damage mechanism w
isdened as
Phf _w k gf f h UDhf _wkUFMS; 1
where k is a given time period, h is the different release hole
sizes.Four values for h are here considered: 1=4'', 1'', 4'' and
16'', whichcorrespond to small, medium, large and rupture,
respectively,
gf f h generic failure frequency for a given equipment and
holesize h and is obtained from a representative failure database,
andgf f total hgf f h, Dhf _wk damage factor related to the
applicabledamage mechanism w and it modies the gf f h to make it
specicto the equipment under evaluation, FMS is management
systemfactor that accounts for the inuence of the facility's
managementsystem on the mechanical integrity of the plant and is
oftenobtained by the application of a questionnaire, through which
ascore in 0;1000 is measured. Then, FMS 100:02:pscore1, wherepscore
0:1score.
Thus, Phf _w k is computed for a given type of damage mechan-ism
w and for the different hole sizes h. As it can be seen in Eq.
(1),
Dhf _wk is the only factor that makes Phf _w k vary over time,
whilegf f h and FMS are kept constant. It is assumed here that
theequipment of interest is subject to both internal (w1) and
external(w2) corrosion causing thinning.
2.2. Damage factor, inspection effectiveness and Bayesian
updating
The damage factor Dhf _wk is computed with available
datagathered from inspection. Fig. 1 (Step 1Step 6) shows a
owdiagram for determination of the thinning damage factor.
Note,however, that inspection programs (the combination of
Non-Destructive Evaluation (NDE) methods such as visual,
ultrasonicand/or hydrostatic test, used to determine the
equipment
Step 2: Determine the time-in-service( ) since the last
inspection reading.age
Step 5: Determine damage factor usingTable 2.
Step 3: Determine the corrosion rate .r
Step 4: Determine using Equation (2).Art
Step 1: Determine the number ofinspections and the
correspondinginspection effectiveness categoryfor all past
inspections. For all pastinspections, combine inspections tothe
highest effectiveness performed.
Step 6:
Determine the adjustment factors:- Online Monitoring-
Injection/Mix Points- Dead Legs- Welded Construction- Maintenance-
Settlement
Determine damage factor forthinning using Equation (3).
Step 7: Update the degree ofconfidence by using Equation (4)
andestimate the probability of failure atfor the hole size via
Equation (5).k h
Obs: Note that thedamage factor is computed
for the three damage states 1 ( ),2 (2 ) and 3 (4 ).
rr r
Step 8: Compute the total probabilityof failure at time
through
Equation (6).k
Fig. 1. Determination of the probability of failure.
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265 255
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condition) vary in their effectiveness for locating and
sizingdamage, and therefore for determining corrosion rates and
therespective damage factor. These limitations may result from
theinability to cover 100% of the areas subject to damage,
inappropri-ate training of personnel as well as from inherent
inadequacy ofsome test methods to identify and estimate damage
[15].
In this way, the corrosion rate r, which is determined
byinspection, is uncertain, and such an uncertainty is propagatedto
Dhf _wk and Phf _w k . Therefore, it is recommended to
useinspection techniques of adequate effectiveness to increase
thedegree of condence on the observed r. Table 1 classies
theinspection effectiveness into ve categories from highly
effective(A) to ineffective (E). In fact, the more thorough the
inspectionprogram, the smaller the uncertainty on r.
However, decision makers may not assign inspection plans
onlycomposed of the highest effective techniques available because
ofbudget constraints. In fact, they combine over equipment's life
low-cost/low-effective inspections with higher effective, but more
expen-sive inspections. In Fig. 1 Step 1, note that we need to
determine thenumber of all past inspections and their corresponding
effectivenesscategory to estimate Dhf _wk. It is also possible to
combine inspectionsto the highest effectiveness performed as 2B 1A,
2C 1B and2D 1C. Note that these rules are not applied to category
E.
Next, in Fig. 1 Step 4, we estimate the metal loss parameterArt
, which is determined from age (time in service since the
lastinspection) given in Fig. 1 Step 2, and the corrosion rate r
that isestimated in Fig. 1 Step 3, as follows:
Art max 1 trdr:agetminCallow
; 0:0
; 2
where trd is the thickness reading, tmin is the minimum
requiredwall thickness, Callow is the corrosion allowance.
Then, the damage factor Dhnf _wk (Fig. 1 Step 5) is based on
thenumber of highest effectiveness inspections (Fig. 1 Step 1)
andthe value of Art (Fig. 1 Step 4). Thus, depending on these
twopieces of information, Dhnf _wk is chosen from Table 2.
Moreover, asshown in Fig. 1 Step 6, we may take into account
adjustmentfactors for on-line monitoring FOM , injection/mix points
FIP , deadlegs FDL, welded construction FWD, maintenance FMA and
settle-ment FSM in order to update D
hnf _wk, as follows in Eq. (3):
Dhf _w k Dhnf w k :FIP :FDL:FWD:FMA:FSM
FOM: 3
As we argued previously, the corrosion rate r is a random
variableand because of imperfect inspections, the estimated r may
differfrom the actual corrosion rate. Additionally, even for k 0
(at thebeginning of equipment's life), when useful data may not
beavailable, we need to predict Dhf _wk at a point in the
future.
Given this uncertainty on r, a conservative standpoint isadopted
since it is assumed that it is possible to observe corrosionrates
twice or even four times higher than the one estimatedby the
inspection technique; we call these situations: damagestates 1, 2
and 3 for the cases for which the corrosion rates are r, 2r
and 4r, respectively. Generally, these higher-than-expected
corro-sion rates are localized in some points of equipment, but
theyusually remain undetected until failure occurs [15]. However,
thebetter the quality of reliability data, the lower the chance
ofoccurrence of damage states 2 and 3.
Table 3 shows the degree of condence, 0 r , that the
actualcorrosion rate r falls into these three possible discrete
damagestates i 1; 2; 3; 0 r is assigned based on the source and
qualityof the data available at k 0. Thus, the information given
inTable 3 may be used as a priori degree of condence on
thecorrosion rate r at k 0, which should be updated in order
toconsider the current damage state of the equipment whenevernew
inspections are performed at some time k40 in the future.This
updating step, which measures the impact of the inspectionprogram
on the degree of condence on r (and on the statedamage), may be
done through Bayesian inference [16] as follows:
k rijr Lk rjri Uk1 ri
Lk rjr1 Uk1 r1 Lk rjr2 Uk1 r2 Lk rjr3 Uk1 r3 ;
4where i 1; 2; 3 is the damage state, k 1; ; m is the time
step,m is the number of time steps considered in the planning
horizon, ris the observed corrosion rate estimated from inspection,
r1 r,r2 2r and r3 4r is the corrosion rates related to damage
states 1,2 and 3, respectively, k1 ri is the prior probability of
the damagestate i at time k, Lk rjri is the likelihood of observing
the result r ofan inspection performed at k given that the
equipment item isunder the damage state i, k rijr is the posterior
distribution for thedamage state i. Note that k rijr becomes the
prior distributionwhen the next inspection takes place, which
permits the Bayesianupdating of the degree of condence on r.
The likelihood Lk rjri depends on the effectiveness of
theinspection technique. Indeed, Table 4 quantitatively expresses
thisclassication as the likelihood that the observed damage
state(collected from an inspection program) actually represents
thetrue state. Thus, Eq. (4) provides a manner to update the degree
ofcondence based on the inspection effectiveness. In this way, it
isexpected that the knowledge acquired from the inspection pro-gram
reduces the uncertainty about the actual deterioration stateof the
equipment, and this information is used to bring up-to-datethe
corrosion rate r, the damage factor Dhf _wk, and then Phf _wk.
In fact, there will exist a different damage factor Dhf _wk; ri
foreach damage state 1 r1 r), 2 (r2 2r) and 3 (r3 4r) as it
isindicated in a remark in Fig. 1. Then, by modifying Eq. (1),
itfollows that
Phf _w k; ri gf f h UDhf _w k; ri UFMS:
Then, Phf _w k is estimated for the hole size h as follows:
Phf _w k 3
i 1Phf _w k; ri k ri ; for k 0;
Phf _w k 3
i 1Phf _w k; ri k rijr ; otherwise: 5
Table 1Inspection effectiveness categories.Source: Ref. [11]
Qualitative inspectioneffectiveness category
Description
(A) Highly effective The inspection methods will correctly
identify the true damage state in nearly every case (or 80100%
condence).(B) Usually effective The inspection methods will
correctly identify the true damage state most of the time (or 6080%
condence).(C) Fairly effective The inspection methods will
correctly identify the true damage state about half of the time (or
4060% condence).(D) Poorly effective The inspection methods will
provide little information to correctly identify the true damage
state (or 2040% condence).(E) Ineffective The inspection method
will provide no or almost no information that will correctly
identify the true damage state and are
considered ineffective for detecting the specic damage mechanism
(less than 20% condence).
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265256
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Table 2Thinning damage factors.Source: Ref. [11]
Art Inspection effectiveness
E 1 Inspection 2 Inspections 3 Inspections
D C B A D C B A D C B A
0.02 1 1 1 1 1 1 1 1 1 1 1 1 10.04 1 1 1 1 1 1 1 1 1 1 1 1 10.06
1 1 1 1 1 1 1 1 1 1 1 1 10.08 1 1 1 1 1 1 1 1 1 1 1 1 10.1 2 2 1 1
1 1 1 1 1 1 1 1 10.12 6 5 3 2 1 4 2 1 1 3 1 1 10.14 20 17 10 6 1 13
6 1 1 10 3 1 10.16 90 70 50 20 3 50 20 4 1 40 10 1 10.18 250 200
130 70 7 170 70 10 1 130 35 3 10.2 400 300 210 110 15 290 120 20 1
260 60 5 10.25 520 450 290 150 20 350 170 30 2 240 80 6 10.3 650
550 400 200 30 400 200 40 4 320 110 9 20.35 750 650 550 300 80 600
300 80 10 540 150 20 50.4 900 800 700 400 130 700 400 120 30 600
200 50 100.45 1050 900 810 500 200 800 500 160 40 700 270 60 200.5
1200 1100 970 600 270 1000 600 200 60 900 360 80 400.55 1350 1200
1130 700 350 1100 750 300 100 1000 500 130 900.6 1500 1400 1250 850
500 1300 900 400 230 1200 620 250 2100.65 1900 1700 1400 1000 700
1600 1105 670 530 1300 880 550 500
0.02 1 1 1 1 1 1 1 1 1 1 1 1 10.04 1 1 1 1 1 1 1 1 1 1 1 1 10.06
1 1 1 1 1 1 1 1 1 1 1 1 10.08 1 1 1 1 1 1 1 1 1 1 1 1 10.1 2 1 1 1
1 1 1 1 1 1 1 1 10.12 6 2 1 1 1 2 1 1 1 1 1 1 10.14 20 7 2 1 1 5 1
1 1 4 1 1 10.16 90 30 5 1 1 20 2 1 1 14 1 1 10.18 250 100 15 1 1 70
7 1 1 50 3 1 10.2 400 180 20 2 1 120 10 1 1 100 6 1 10.25 520 200
30 2 1 150 15 2 1 120 7 1 10.3 650 240 50 4 2 180 25 3 2 150 10 2
20.35 750 440 90 10 4 350 70 6 4 280 40 5 40.4 900 500 140 20 8 400
110 10 8 350 90 9 80.45 1050 600 200 30 15 500 160 20 15 400 130 20
150.5 1200 800 270 50 40 700 210 40 40 600 180 40 400.55 1350 900
350 100 90 800 260 90 90 700 240 90 900.6 1500 1000 450 220 210 900
360 210 210 800 300 210 2100.65 1900 1200 700 530 500 1100 640 500
500 1000 600 500 500
Table 3Condence in predicted damage rate.Source: Ref. [15]
Damage state category Example-general corrosion Actualdamagerate
range
Low-reliabilitydata
Moderate-reliabilitydata
High-reliability data
Damage State 1: The damage in the equipmentis no worse than what
is expected based ondamage rate models or experience.
The rate of general corrosion is less than orequal to the rate
predicted by past inspectionrecords, or historical data if no
inspectionshave been performed.
Predictedrate (r) orless
0.50 0.70 0.80
Damage State 2: The damage in the equipmentis somewhat worse
than anticipated. Thislevel of damage is sometimes seen in
similarequipment items.
The rate of general corrosion is as much astwice the predicted
rate.
Predictedrate (r) totwo timesrate (2r)
0.30 0.20 0.15
Damage State 3: The damage in the equipmentis considerably worse
than anticipated. Thislevel of damage is rarely seen in
similarequipment items, but has been observed onoccasion
industry-wide.
The rate of general corrosion is as much asfour times the
predicted rate.
Two (2r) tofour times(4r)predictedrate
0.20 0.10 0.05
Examples (a) Publisheddata
(a) Laboratorytesting
(a) Field data
(b) Corrosionrate tables
(b) Coupon datareecting ve or moreyears of experiencewith the
processequipment
(b) Limitedcouponcorrosiontesting
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265 257
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First part of Eq. (5) allows propagating the uncertainty on the
statedamage to Phf _w k at k 0, when just information given in
Table 3is available. As an inspection technique is performed at
k40, weupdate the degree of condence on the state damage based
oninformation in Table 4 and via Eq. (4), which yields k rijr ,
andthen we estimate Phf _w k . In this way, the effectiveness of
theinspection techniques performed at each period k, which
isdetermined in the adopted inspection program, directly
inuencesthe computation of Phf _w k . Thus, Eq. (5) provides (Fig.
1 Step 7)the up-to-date Phf _w k for one damage mechanism w, for
whichthe total probability is given in Fig. 1 Step 8 as
follows:
Pf _w k hPhf _w k : 6
However, note that the procedure of Fig. 1 should be
repeatedtwice as the equipment of interest is subject to internal
(w1 IC)and external (w2EC) corrosion. Thus, Pf _wi k (i 1; 2)
should beestimated for both types by considering the internal (rw1
) andexternal (rw2 ) corrosion rates. Then, by assuming w1 and w2
areindependent mechanisms, it follows that
Pf k Pf _w1 k Pf _w2 k Pf _w1 k :Pf _w2 k : 7
2.3. Consequence of failure
In accordance with the API-RBI methodology, the consequencesof
failures may be expressed in nancial terms (FC). According to[11],
the steps of a consequence analysis for a xed pressurizedequipment
are as follows:
determine the representative uid; select a set of release hole
sizes; calculate theoretical release rate/mass; estimate uid
inventory; establish release type; estimate impact of detection and
isolation systems; calculate adjusted release rate/mass; determine
ammable/explosive consequences; determine toxic consequences;
determine non-ammable and non-toxic consequences; determine
component damage and personnel injury conse-quence areas (CA);
determine FC.
The underlying idea of a level 1 consequence analysis is
toestimate the consequences of releases of hazardous uids at
thehole size h level for which there are several costs to take
intoaccount. First, the API-RBI methodology recommends that
weconsider the costs of repair and replacement (FCrep) of
damaged
equipment, which are calculated as
FCrep hgf f
h:holecosth
gf f total
!:matcost;
where holecosth is the equipment repair costs ($) based on
carbonsteel prices and matcost is a material cost factor to adjust
FCrep toother materials. Costs of damage to surrounding equipment
inaffected areas (FCaf f a) should also be included if the failure
resultsin a ammable (or explosive) event. In fact, these costs are
given as
FCaf f a CAcmd:equipcost;where CAcmd is the nal damage
consequence area (m2) andequipcost is the process unit replacement
cost for components($=m2).
Costs associated with production losses and business
interrup-tion as a result of downtime for repair or replacement of
damagedequipment (FCprod) are also considered. These costs are
deter-mined based on the amount of downtime associated with
thespecic equipment that were subject to a loss of containment
andall equipment also affected by the release in the consequence
area.Then, this cost is given as
FCprod OutagecmdOutageaf f a
:prodcost;
where Outagecmd hgf f h:Outageh=gf f total
:Outagemult is the
weighted (on release hole size) number of days of
downtimerequired to repair the specic piece of equipment that is
beingevaluated (days), Outageh is the number of downtime days
torepair damage associated with the h release hole size
(days),Outagemult is the equipment outage multiplier, Outageaf f a
numberof days of downtime required to repair damage to the
surroundingequipment (days), prodcost cost of lost production due
to down-time to repair equipment ($=days).
Other costs to consider are those due to potential casualties
ofpersonnel and/or of communities in the vicinity, which are given
as
FCinj CAinj:poddens:injcost;
where CAinj is the nal personnel injury consequence area
(m2),poddens is the population density of personnel or employees in
theunit (personnel=m2), and injcost is the cost associated with
seriousinjury of fatality of personnel ($). Finally, the
environmental cleanup
costs (FCenv) are also estimated as FCenv hgf f h:volhenv=gf f
total
:
envcost ,where envcost is the environmental clean-up costs
($=barrel),
volhenv is the spill volume in barrels to be cleaned up
calculated foreach of the 4 release hole sizes selected. The
fraction of the released
liquid pool that evaporates is also needed to estimate
volhenv.Then, FC may be estimated adding up all abovementioned
individual costs as in Eq. (8):
FC FCrepFCaf f aFCprodFCinjFCenv: 8Even though the consequence
of failure FC needs to be evaluatedonly once for a particular
equipment, this step of a quantitative RBIassessment is still very
demanding. First, because of the largeamount of information
requirements necessary to feed the equa-tions previously presented
even in a level 1 consequence analysis.Indeed, note that these
expressions are given in a summarizedformat since they encapsulate
other pieces of information. Forexample, volhenv depends on the
fraction of material that willevaporate, the uid liquid density,
and the normal boiling point.Moreover, giving the same level of
attention to all equipment ispractically impossible due to the huge
number of components inreneries and petrochemical industries; even
after prioritizingequipment based on a preliminary qualitative risk
analysis, thenumber of components to be evaluated quantitatively is
stillvery large.
Table 4General corrosioninspection effectiveness.Source: Ref.
[15]
Damage ratestate
Range of factual damage rate Likelihood thatinspection
resultdetermines the truedamage
A B C D/E
1 Measured rate or less 0.90 0.70 0.50 0.332 Measured rate to 2
measured
rate0.09 0.20 0.30 0.33
3 24 measured rate 0.01 0.10 0.20 0.33
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133 (2015) 253265258
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Note that, in order to obtain efcient inspection programs,
theproposed RBI multi-objective framework does not require
thecalculation of FC. In fact, since FC is kept constant over
time(API-RBI methodology [11]), only the damage factor needs to
bere-computed at every k, which then allows updating Pf k .
Thus,this feature results in a considerable reduction of
effort.
3. Problem statement and formulation
Let an inspection program x be represented by the
followingvector:
x x1;1; ; x1;k; ; x1;m; ; xj;1; ; xj;k; ; xj;m; ;xn;1; ; xn;k; ;
xn;m; 9
where n is the number of available NDE inspection techniques
andm is the number of time steps considered in the planning
horizon.Each element xj;k of x is either 0 or 1, j 1;;n and k 1;;m.
Ifxj;k 1, then the inspection technique j is performed at k.
Other-wise, there is no inspection of type j at k. In this way, at
period k, atmost one inspection using technique j is allowed.
Notice thatvarious inspection methods are allowed in the same
period k, asthey may provide different information about the actual
state ofequipment. In this case, however, we apply the Bayesian
updatinggiven in Eq. (4) considering the inspection technique of
highesteffectiveness.
The multi-objective optimization problem, which consists
ofselecting efcient inspection programs x that represent the
com-promise between C x and Pf x is formally dened as follows:
min C x n
j 1cj
m
k 1xj;k
!cp
n
j 1m
k 1xj;kcd
jA J'm
k 1xj;k; 10
min Pf x m
k 1Pf k ; 11
s:t: xj;1xj; tj; max1Z1; 8 j: 12
The inspection cost objective function (Eq. (10)) is related to
theinspection activity, where cj is the cost of performing
inspectionusing technique j, cp is the cost of qualied personnel
perinspection, cd is the downtime cost per period, and J' is the
setof intrusive inspection techniques that require entry into
theequipment, causing downtime. Eq. (11) corresponds to the totalPf
x associated with an inspection program x within the
planninghorizon m; it is the sum of Pf k that comes from Eq. (7).
Note thatPf k directly depends on the damage factor and on the
effective-ness of the inspections that compose program x, as
explained inSection 2.2.
Yet, international standards and/or local government
regula-tions (such as the Brazilian Regulation Standard ([17])
generallyestablish that an inspection technique j should be
performed atleast once within a given time period. In this way, it
is possible todene a maximum allowed interval (tj; max, j 1;;n)
betweenconsecutive inspections. By taking this into account, Eq.
(12) iswritten, where represents a period with an inspection (xj;
1)and can possibly assume any of the values 1;;mtj; max1.Thus, the
constraints in Eq. (12) concern the summations ofinspections over
groups of tj; max1 periods, which have to be atleast 1, i.e., at
least one inspection technique j has to be performedin such an
interval.
A risk constraint, i.e. RxrRt or equivalently Pf x rRt=FC,could
be added to the multi-objective formulations (10)(12). Inthis case,
the knowledge of FC would be required a priori and allthe obtained
solutions (inspection plans) would presentPf x rRt=FC. Indeed, such
solutions are a subset of the set ofinspection plans resulting from
the problems (10)(12) as it is
presented, with no upper limiting value for R x . Therefore,
theproposed approach is general, allowing the consideration of a
riskconstraint in a posterior step, as soon as FC has been
computed.Thus, the identication of the valid inspection plans is a
simpletask as they would be in the region of the Pareto front dened
bythe risk constraint. This characteristic will be shown in
theapplication example in Section 6.
4. Proposed Multi-Objective Genetic Algorithm
In a multi-objective optimization, a set of
non-dominatedsolutions is obtained representing the compromise
among objec-tives. Thus, the concept of (non-)dominance relation
plays acentral role in multi-objective techniques. A solution is
non-dominated if, for all objectives, it has a performance at least
asgood as that of the other solutions and, at least for one of
theobjectives, its performance overcomes that of the others. On
theother hand, the dominance relation is mathematically dened
asfollows:
x1gx23 f i x1 r f i x2 ; 8 i and f i x1 o f ix2 for some i;
13where g denotes dominance, x1 is a non-dominated solution fora
minimization problem and x2 is a dominated solution for thesame
problem; f i denotes the ith objective function (e.g.,C x ; Pf x).
If one of the conditions on the right-hand side ofEq. (13) is not
satised, both x1 and x2 are said to be non-dominated. That is, for
a number of objectives, x1 overcomes theperformance of x2 and, for
the remaining objectives, x2 overcomesthe performance of x1. The
non-dominance relation is alsoobserved when x1 x2.
In this section, the proposed MOGA is tailored for
solvingproblems (10)(12). First, one of the characteristic of this
approachis that the genetic operators provide only feasible
inspectionprograms as outcomes. This feature has the following
advantages:(i) the search space to be explored by the MOGA is
reduced; (ii) theMOGA is prevented from getting stuck into an
unfeasible part ofthe search space; (iii) unnecessary tness
evaluations of unfeasibleindividuals are avoided; and (iv) the use
of penalty functions dueto unfeasibility is not required.
For the problem characterized in Section 3, the total number
ofsolutions in the search space, when the constraints in Eq. (12)
arenot taken into account, is given by 2n:m. The number of
feasibleinspection plans for a given technique j is am;pj xm f pj
x, whichis dened as the coefcient of xm in the x-power series
expansionof f pj x, where
f pj x 2 xx2xpj xpj1xx2xpj ;
for pj tj;max1. Hence, the percentage of the search
spaceconcerning feasible inspection plans is given by
am;p1 Uam;p2 UUam;pn2m Un
100%:
As an illustration, suppose n 1, m 20, p 3 (thus, t1;max 2).The
x-power series expansion of f pj x is2x4x27x313x424x5
44x681x7149x8274x9504x10927x11
1;705x123;136x135;768x1410;609x1519;513x16
35;890x1766;012x18
121;415x19223;317x20:Therefore, the number of feasible
inspection programs isa20;3 223;317 and it represents about 21:3%
of the search space.
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265 259
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For further details on enumeration techniques, the
interestedreader is referred to [18].
The next sections detail the individual representation alongwith
the proposed genetic operators for the generation of
initialpopulation, crossover and mutation. Recursive algorithms
aredeveloped for the elaboration of inspection plans.
Feasibilityevaluation of xj;1, , xj; tj; max1 is required whenever
aninspection takes place (i.e., xj; 1) and for every techniquej
1;;n. The selection and update of the auxiliary populationPaux are
the same as those presented in Ref. [19]. The selection isbased on
the (non-)dominance relation among solutions. Domi-nated solutions
are eliminated from the current population andthe non-dominated
ones are stored in Paux. In order to recover thenumber of
individuals N in population, solutions from Paux arerandomly
chosen. Paux is updated at every iteration of the algo-rithm. For
further details, the reader is referred to [19].
4.1. Individual representation
An individual is represented by the vector in Eq. (9),
whoseentries are either 1 or 0 if an inspection using the related
techniqueis either or not performed in the associated period,
respectively.Here, an integer representation of individuals is
adopted.
For the sake of illustration, consider n 3, m 6, t1;max 2,t2;max
3, t3;max 4 and the inspection plan in Table 5, indicatingan
inspection plan such that: (i) the rst technique should beperformed
at periods 2 and 5; (ii) the second technique should beperformed at
periods 1, 3 and 4; and (iii) the third techniqueshould be
performed at periods 1 and 6.
4.2. Generation of initial population
Each of the N individuals of the initial population is
randomlygenerated according to a discrete uniform distribution with
addi-tional features to handle unfeasibility, which are presented
in thealgorithm of Fig. 2. The underlying idea of the algorithm is
that, fora given technique j, every time an inspection is
established(xj; 1), a new group of values for the future periods
xj;1,,xj; tj; max1 is generated for feasibility investigation. In
this way,an inspection to be carried out at period requires the
restart ofthe algorithm from 1.
Indeed, groups of tj;max1 periods, starting from 1,
areconsidered one at a time. The values xj;b, b 1;;tj;max1are
randomly set either to 0 or to 1, and whenever xj;b 1, arecursion
of the algorithm starting from b1 is required. If all xj;bare equal
to 0, then no inspection is performed within theconsidered period
(S 0), and the constraint in Eq. (12) is violated.In order to
tackle the unfeasibility, a position p among1;; tj;max1 is selected
and the corresponding value isset to 1; then, a recursion with p as
argument is called. These stepsare repeated until the nal period m
is reached. Once thisreasoning is applied for each of the n
considered techniques, afeasible individual is generated.
4.3. Crossover and replacement
Since the values of xj;b are either 0 or 1, the usual
binarycrossover [20] is performed between two individuals (parents,
e.g.,
x1 and x2). Such an operator is combined to a procedure devised
toevaluate the offspring feasibility and to transform
unfeasibleindividuals that might have been created into feasible
ones. Theparents' positions are interchanged at randomly chosen cut
points(c) so that to generate two new individuals: child 1 and
child2 that are, respectively, the modied x1 and x2, since the
replace-ment is automatically performed (children replace parents).
Fig. 3adepicts the crossover between two individuals when n 3, m
6,c 4, t1;max 2, t2;max 3, and t3;max 4. Then, the investigationand
handling of an eventual unfeasibility for each new individual,per
technique, take place.
The algorithm used to perform these tasks is essentiallythe same
as in Fig. 2; the only exception is step 1(b), given thatin the
crossover the values xj;b are not created. Note that for
theillustrated example, the crossover procedure generated an
unfea-sible offspring (child 2) that violated the maximum number
ofperiods without an inspection using technique 1. In Fig. 3b,
theunfeasibility is identied and a possible solution is given. As
anoutcome, child 2 becomes feasible.
4.4. Mutation
As in the crossover, given that either xj;k 0 or xj;k 1, 8 j
and8k, the binary mutation [20] combined to a mechanism to
rendereventual unfeasible individuals into feasible ones is
applied. Forevery position, a uniform random number uA 0;1 is
generated;given a position j; k, if u is less than or equal to the
mutation
Table 5Individual representation for MOGA (inspection plan).
Technique 1 Technique 2 Technique 3
Period 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6Inspection plan 0 1 0
0 1 0 1 0 1 1 0 0 1 0 0 0 0 1
Fig. 2. Generation of a feasible plan for a given technique.
Fig. 3. (a) Example of binary crossover procedure and (b)
solving unfeasibility ofchild 2.
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265260
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probability (pm), the value of xj;k is changed either (i) from 0
to 1 or(ii) from 1 to 0; otherwise, xj;k remains the same.
Note that only mutations of type (ii) generate
unfeasibility,since additional inspections due to type (i)
mutations do not harmthe individual's feasibility. In this way,
whenever a mutation oftype (ii) occurs, the related technique and
period are stored invectors it and ip, respectively. Once all
positions of an individualhave been submitted to the binary
mutation procedure, it and ipare of the same length (jitj jipj). If
it and ip are both empty,which means no mutations of type (ii) have
happened, the relatedindividual is still feasible.
On the other hand, if jitj jipj40, it is necessary to
investigateeventual unfeasibility arising due to type (ii)
mutations. If techni-que j and period k are, respectively, at the
same positions of it andip and the corresponding mutation resulted
in an unfeasibility(a greater number of periods without inspections
using technique jthan permitted), the xj;k value is restored to 1
as if no mutation hadtaken place at position j; k. Otherwise, if a
mutation of type(ii) has occurred but has not generated an
unfeasibility, the productof the mutation remains unchanged, i.e.
xj;k 0. The idea is to modifyindividuals as least as possible after
the mutation operator has beenapplied in order to preserve the MOGA
evolution trend.
The algorithm of Fig. 4 summarizes the investigation
procedurefor unfeasibility over an individual for a given technique
as well astheir associated treatment in order to render the
inspection planfeasible. Note that steps until (iii) are basically
the same as inalgorithm of Fig. 2. These steps are necessary
because of altera-tions in feasibility analysis due to possible 1's
provided bymutations of type (i), which also demand the recursion
of thealgorithm starting from the immediate subsequent position.
Theelements of it and ip that are eliminated in steps 4 and 5 are
thoseinvolving already solved unfeasibility. Thus, at the end,
ifjitj jipj40, the remaining elements refer to positions that
havebeen submitted to type (ii) mutations but have not
generatedunfeasibility.
4.5. Overview of the proposed RBIMOGA framework
The proposed approach couples the RBI methodology describedin
Section 2 to the MOGA-based optimization procedure pre-viously
presented, which entails constraints to comply withpossible
existing international standards and/or regulations. Anoverview of
RBIMOGA is provided in the owchart of Fig. 5.
5. Numerical experiments
In this section, a numerical example is solved by the
proposedRBIMOGA approach in order to compare the obtained
resultsagainst the real non-dominated inspection plans, which are
in turnprovided by an exhaustive recursive algorithm. A planning
horizonof m 10 years is considered, n 3 inspection techniques
areavailable, and two damage mechanisms act on the
equipment(internal and external corrosion) both causing thinning.
Techni-ques 2 and 3 require equipment shutdown; thus they
generatedowntime costs. This example has 113,092,992 feasible
inspectionplans, which represent 10.53% of the entire search space,
and 46non-dominated solutions.
The required parameters for computing C x (Eq. (10)) and Pf x
(Eq. (11)) are presented in Table 6; consider also that
low-reliability data (see Table 3) was available at k 0 to
estimatethe corrosion rates. The gf f hfor h1/4, 1, 4 and 16
are,respectively, 8 106; 2 105; 2 106 and 6 107 fail-ures per year
and FMS 1. Table 7 presents the MOGA parameters.
In order to evaluate the stochastic behavior of RBIMOGA, weran
100 replications. Some descriptive statistics are shown inTable 8.
In all cases, RBIMOGA was able to nd almost allsolutions from the
true Pareto front. In the worst case, about91% of the exact
solutions were provided by RBIMOGA. Such aproportion is
approximately 98% for the best case. The exhaustiveFig. 4.
Evaluation and solution of eventual unfeasibility after
mutation.
MOGA+RBI
Selection andupdatePaux
Mutation
No
Yes
Replacement
Crossover
End
Start
Generation ofinitial population
Obtain ,= 1/4, 1, 4, 16,and via RBI
gffh
F
h
MS
Update Paux
Maximumnumber of generations is
reached?
Fitnesse
valuation
iN
=1,...,
Generationnumber =
0
C x( )i
P xf ( ) via RBI(Sections 2.1 and 2.2)
i
Fig. 5. Proposed RBIMOGA.
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265 261
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recursive algorithm required 33 minutes to return the true
Paretofront in a Windows operating system, Intels CoreTM i7 2.20
GHzprocessor and 8 GB of RAM, while each run of RBIMOGA tookonly
about 2.5 s, which is around 0.13% of the time needed by
theexhaustive algorithm. Fig. 6 depicts the true Pareto front
alongwith the worst and best cases. It can be observed that even
theworst RBIMOGA front is near the true one.
6. Application example: oil and gas separator vessel
In this section, the proposed RBIMOGA is applied to
obtainnon-dominated inspection programs for a separator vessel of
oiland gas by considering n 3 inspection techniques (1:
externalexam; 2: internal exam; 3: hydrostatic test) and a horizon
ofm 20 years. This equipment item is exposed to internal
andexternal corrosions that cause thinning. It is here considered
thatexternal exams are ineffective (E) to evaluate internal
corrosion,while they are usually effective (B) to measure thinning
due toexternal corrosion. Internal exams are treated as usually
effective(B) for internal and external corrosions, and the
hydrostatic test ishighly effective (A) for both damage mechanisms.
Both internalexam and hydrostatic test are intrusive techniques;
thus theygenerate downtime costs.
The Brazilian Regulation Standard ([17]) establishes
maximumtimes to perform at least one inspection for these three
techni-ques: 4, 8 and 16 years, respectively. Thus, t1;max 3,
t2;max 7, andt3;max 15 years; it was considered a Type II vessel
and a plant thathas its Own Service of Inspection (see Ref. [17]).
The same gf f h
and FMS values used in the previous example are here
adopted.
Inspection costs along with the characteristics of the
damagemechanisms, which are used to calculate C x , Art , Dhf _wk,
Pf k and Pf x , are summarized in Table 9. The MOGA parameters
arethe same as in Table 7, but the number of cut points is c 15.
Forthis problem setting, the number of feasible inspection
programsis in the order of 1017 and it represents 50.77% of the
entire searchspace. The exhaustive evaluation of these feasible
programs isimpossible in practice. Then, we need to resort to the
MOGAalgorithm developed in Section 4.
For this application, the plan with inspection activities
per-formed as late as possible in accordance with [17] (see Table
10)was forced to be in the initial population so as to guarantee
that itwould be evaluated by RBIMOGA. The gray cells in Table
10represent that an inspection is to be performed in the
associatedperiod (column) using a specic technique (row). Note that
suchan inspection program is not the only one suggested by [17],
but itis the rst that comes to mind, as it would be, at rst glance,
thecheapest one that still complies with this particular
regulationstandard.
For this example, RBIMOGAwas replicated 30 times and eachrun
required approximately 5 s to be performed. All fronts weresimilar
and some descriptive statistics are shown in Table 11.However,
instead of using one of these 30 fronts, an evaluation ofthe (non)
dominance relation among the solutions of all fronts wasperformed.
The resulting Pareto front contains 122 solutions and ispresented
in Fig. 7.
The inspection programs (IPi; i 1; 2; 3) associated with
thesolutions indicated in Fig. 7 are presented in Table 12. If
solutionIP1 and the one presented in Table 10 are compared, one
mayconclude that in spite of having the same inspection cost for
the20 years, a slight modication in the schedule of techniques1 and
2 resulted in a smaller Pf x for solution IP1. Thus, theinspection
plan of Table 10 is dominated by solution IP1.
For the sake of illustration, Fig. 8 exemplies how Pf k
forsolution IP1 varies over time because of the damage factor
andbased on the Bayesian updating procedure given in Section 2.
First,note that Pf k is not necessarily a monotone function.
This
Table 6RBI parameters and inspection costs, validation
example.
Technique tj; max (years) cj Damage mechanism Thickness (mm)
Internal corrosion External corrosion Initial Minimum
(tmin)Effectiveness
1 1 1000 E B 12 102 3 5000 B B3 7 10,000 A A
cp cd Corrosion rate r (mm/year) Corrosion allowance Callow
(mm)100 500 rIC 0:4 rEC 0:6 1FOM FIP FDL FWD FMA FSM 1
Table 7MOGA parameters.
Parameter Value
Population size (N) 250Number of generations (Ngen)
500Probability of crossover (pc) 0.95Number of cut points (c)
8Probability of mutation (pm) 0.05
Table 8Results for validation example: 100 replications of
RBIMOGA.
Statistics Number of solutions Number of exact solutions
Minimum 43 42Median 46 44Maximum 46 45Mean 45.57 44.28Std. dev.
0.5730 0.5333
64 108
0.00
0.02
0.04
0.06
0.08
C(x) (in 104)P
f(x)
Fig. 6. Exact and simulated Pareto fronts for validation
example.
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265262
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happens because the relatively low-cost/low-effective
inspectiontechniques 1 and 2 are performed in earlier steps (ko16
ofequipment's life, whereas the high-cost/high-effective
inspectiontechnique 3 is just adopted at k 16. Indeed, for this
case, despitethe expected better condition of the equipment at
ko16, theuncertainty on the corrosion rate is then high since
little helpfulinformation is available at that period. For
instance, consider Pf k at k 12 and the dip at k 16 in Fig. 8. Note
that even though amore severe corrosion rate would be expected at k
16, moreconservative estimates for Pf k are provided at k 12
becausemore condence is put on state damages 2 and 3 at k 12
sinceless relevant information is available. On the other hand, as
aninspection with higher effectiveness is carried out at k 16
onehas more condence in the observed corrosion rate r1 related
tostate damage 1.
A return on investment (ROI) analysis can aid the decision
forthe adoption and implementation of an inspection plan from
the
Pareto front:
ROI Pf xj
Pf xi C xj C xi ;
where i and j are different solutions from the Pareto front.
Forexample, each monetary unit invested in inspection activities to
gofrom solution IP1 to IP2 corresponds to a reduction of 3:43 106in
Pf x . On the other hand, from solution IP2 to IP3 each
additionalmonetary unit would reduce Pf x by 2:94 107. Thus,
highinvestments in inspection do not necessarily mean
signicantreductions on the total Pf x .
The obtained Pf x can be used to compute the risk R x associated
with every inspection program x from the Pareto frontas the nancial
consequence FC is available. In order to illustrate, aconsequence
analysis was performed according to Ref. [11] forthe oil and gas
separator vessel and FC $ 2;246;908:17 wasestimated as given in
Section 2.3. The Pareto front presentingthe trade-off between C x
and R x is given in Fig. 9. Note that thefronts in Figs. 7 and 9
have the same shape and, as expected, theonly difference between
them is the scale factor FC, which isreected in the values of the
vertical axis.
If the risk must be smaller than a preset target value Rt
,possibly to comply with safety regulations, i.e., RxrRt
orequivalently Pf x rRt=FC, Pareto solutions satisfying such a
con-straint are in the region below the corresponding horizontal
linedened by Rt or by Rt=FC in the graphs C x vs. Rx or C x vs. Pf
x ,respectively. As an illustration, for the present example,
ifRxrRt 45;000 and consequently Pf x r0:02, there are 87valid
solutions (all of them are under the horizontal lines shownin Figs.
7 and 9).
It is important to emphasize that in order to obtain the
mostefcient inspection programs regarding both C x and R x
throughthe proposed RBIMOGA approach, it is not necessary to
developthe very demanding task of assessing the consequences of
failuresince the only component that makes the risk level vary over
timeis Pf x . Thus, the effort to accomplish the analysis is fairly
reduced.Even if a risk constraint is mandatory, it can be
considered in aposterior step, when FC becomes available. This is
possiblebecause, as Pf x is an objective to be minimized, such a
constraint
Table 9RBI parameters and inspection costs, application
example
Technique tj; max (years) cj Damage mechanism Thickness (mm)
Internal corrosion External corrosion Initial Minimum
(tmin)Effectiveness
1 3 1000 E B 16 122 7 5000 B B3 15 10,000 A A
cp cd Corrosion rate r (mm/year) Corrosion allowance Callow
(mm)300 1000 rIC 0:200 rEC 0:454 1FOM FIP FDL FWD FMA FSM 1
Table 10Plan with inspections performed as late as possible in
accordance with [17].
Table 11Descriptive statistics for the number of solutions per
front (30 replications).
Minimum Median Maximum Mean Std. dev.
112 116 121 116.4 2.2664
50000 100000 1500000.00
0.02
0.04
0.06
0.08
0.10
0.12
C(x)
Pf(x
)
IP1
IP2
IP3
Fig. 7. Pareto front for application example and some selected
solutions: C x vs.Pf x
M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265 263
-
is a cut in the obtained Pareto front and the solutions of
interestcan be easily identied.
7. Conclusion
In this paper, an original RBIMOGA methodology was devel-oped in
order to provide the decision maker with efcientinspection programs
in terms of both inspection costs C x andrisk R x (by the direct
minimization of Pf x ). The RBI methodol-ogy was used to assess Pf
x related to the candidate inspectionprograms x provided by
MOGA.
The fact of directly considering Pf x as an objective to
beoptimized has two advantages: (i) rst, the user-dened risk
targetRt was no longer required as established in [11]; (ii) the
step ofestimating the nancial consequences FC of failures may
beskipped, saving a lot of effort and time. Moreover, as C x is
also
taken as an objective, the inspection expenditures become
man-ageable and it was possible to assess the impact of
preventioninvestments on Pf x , and then on R x ; this would not be
possibleif only RBI methodology had been adopted.
In the MOGA portion, the genetic operators were adapted forthe
creation of only feasible inspection programs, which should bein
compliance with restrictions that might be imposed by
inter-national standard and/or local regulations. In this way,
MOGAexplored the search space in a more efcient way, as only
itsfeasible portion had been taken into account.
The proposed RBIMOGA was applied to two examples, one ofthem
involving an oil and gas separator vessel. For these cases,it was
possible to provide information on how the inspectionbudget should
be spent more efciently, which involves deningthe whole inspection
program associated with each pair (Pf x ,C x ). An ROI analysis was
performed on the obtained non-dominated inspection programs and it
could be inferred that highinvestments in inspection do not
necessarily yield a signicantreduction of Pf x (thus, R x ).
Finally, the results suggest that theRBIMOGA with the
post-optimization ROI analysis is an effectivetool to support
efcient decisions related to mechanical integrityof equipment.
Acknowledgments
The rst three authors thank the Brazilian research-fundingagency
Conselho Nacional de Desenvolvimento Cientco e Tecno-lgico (CNPq)
for the nancial support through research grants.
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50000 100000 1500000
50000
150000
250000
C(x)
R(x
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M.d.C. Moura et al. / Reliability Engineering and System Safety
133 (2015) 253265 265
A Multi-Objective Genetic Algorithm for determining efficient
Risk-Based Inspection programsIntroductionRisk-Based
InspectionProbability of failureDamage factor, inspection
effectiveness and Bayesian updatingConsequence of failure
Problem statement and formulationProposed Multi-Objective
Genetic AlgorithmIndividual representationGeneration of initial
populationCrossover and replacementMutationOverview of the proposed
RBI+MOGA framework
Numerical experimentsApplication example: oil and gas separator
vesselConclusionAcknowledgmentsReferences