Integrate fault tree analysis and fuzzy sets in quantitative risk assessment

1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print),
ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 12-20 © IAEME
12
INTEGRATE FAULT TREE ANALYSIS AND FUZZY SETS IN
QUANTITATIVE RISK ASSESSMENT
Rachid Ouache1
, Ali A.J Adham2
, Noor AzlinnaBinti Azizan3
1, 2, 3
Faculty of Technology, University Malaysia Pahang, 26300, Malaysia
ABSTRACT
Quantitative risk assessment is the most important step to judge the results of risk estimation
in the process of decision making to improve level of safety. Quantitative risk assessment hasfaced
big problemwith complexity of engineering systemsin term of reliability. In this study, reliability of
risk Assessment proposed to solve problem of uncertainty based on fuzzysets and fault tree analysis
to precise values of top event. The results demonstrated that the model proposed is the best to solve
problem of uncertainty in quantitative risk assessment.
Keywords: Quantitative risk assessment, Fault tree analysis, Fuzzyset, Reliability.
1. INTRODUCTION
In recent decades the world has witnessed an increase in the number of major accidents and
disasters, where considerable efforts are made to control the safety of industrial systems. Risk
assessment approach are designed primarily to reduce the existing risk inherent in engineering
system to a level considered tolerable and maintain it over time.The method of fault tree analysis
used to study the reliability and dependability of complex systems (Ding et al, 2010;Gupta et al,
2007).A study of system reliability, the probabilities are often considered accurate and fully
determinable. It is also assumed that all the information on the performance of the reliability of the
system and its components is provided (Wang et al, 2009). The Improve of reliability for prolonging
the life of the item based on two steps essential, on the one hand, study reliability issues and on the
other hand, estimate and reduce the failure rate. Two approaches for risk analysis, which can be
qualitatively and quantitatively. The qualitative approach used when there is a source of danger, and
when there are no safeguards against exposure of the hazard, and then there is a possibility of loss or
injury. In complex engineering systems, there are often safeguarded against exposure of hazards for
maximizing the level of safeguards, and minimize the level of risk. The quantitative risk assessment
is the approach concerned with this study. Since quantitative risk analysis involves estimation of the
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degree or probability of loss, risk analysis is fundamentally intertwined with the concept of
probability of occurrence of hazards.This studyproposes FTAuses fuzzy setstoprecisethe values of
top events which help us to precisethe value of risk.
2. FUZZY FAULT TREE ANALYSIS FFTA
Probabilistic safety assessment by FTA was considered as an important tool to evaluate the
systems engineering. Boolean algebras are used to mathematically represent the tree diagramand
calculate the output of every logic gate (Haimes et al, 2004; Epstein &Rauzy 2005; Ericson
2005;Huang, Tonga &Zuo 2004, N. Limnios, 2007). The occurrence probability of the undesired top
event is afunction of the reliability data of primary events, which are also known as basic
events(IAEA 2007; Yang 2007).Fault tree analysis can give two types of results, i.e. qualitative
andquantitative results.
2.1 FAULT TREE ANALYSIS
The FTA is composed of a number of symbols to describe events,Boolean gates, and page
transfers. Transfer event symbols are pointers toindicate sub-tree branches that are used elsewhere in
the tree (Ericson 2005; Vesely et al. 1981).
2.1.1 BOOLEAN ALGEBRA
Boolean algebra is the algebra of fault events used in a fault tree tomathematically represent
the relationship between input and output faultevent of a Boolean gate in the tree. This relationship
describes a situation where anoutput of the gate either fails or not (Haimes 2004;Vesely et al. 1981).
Table.1: Booleanalgebras
Rules Engineering symbolism Mathematical Symbolism
Idempotent law X.X=X
X+X=X
X‫ځ‬ X= X
X‫ڂ‬ X= X
Distributive law X. (Y+Z)= X.Y+X.Z X‫ځ‬ሺY ‫ڂ‬ Zሻ= (X‫ځ‬Y)‫(ڂ‬X‫ځ‬Z)
Commutative law X.Y=Y.X X‫ځ‬Y=Y‫ځ‬X
Absorption law X.(X+Y)=X
X+(X.Y)=X
X‫(ځ‬X‫ڂ‬Y)=X
X‫(ڂ‬X‫ځ‬Y)=X
2.1.2 FAILURE PROBABILITY CALCULATION
The failure probability of an output event for two or more independent inputevents combined
by a Boolean OR gate calculated using Eq.(1). And by a Boolean AND gate calculated using Eq. (2).
ܲሺ‫ܣ‬଴ሻ ൌ 1 െ ෑ ሼ1 െ ܲሺ‫ܣ‬௜ሻሽ
௡
௜ୀଵ
ሺ1ሻ
ܲሺ‫ܣ‬଴ሻ ൌ ෑ ܲሺ‫ܣ‬௜ሻ
௡
௜ୀଵ
ሺ2ሻ
Whereܲሺ‫ܣ‬௜ሻ is the failure probability of the input event‫ܣ‬௜, and n is the number of inputevents
to the Boolean gate.

3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print),
ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 12-20 © IAEME
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2.2 FUZZY SET THEORY
2.2.1 Fuzzy Sets
The utility of fuzzy sets lies in their ability to model uncertain or ambiguous data, Fuzzy Sets
FS is important to observe that there is an intimate connection between Fuzziness and Complexity.
Fuzzy sets provide a means to model the uncertainty associated with vagueness, imprecision, and
lack of information regarding a problem or a plant, etc (Dubois, 1980, Zadeh, 1978).
The soft computingis useful tool for solving problems in many fields.It is a high-performance
language for technical computing (Dubois, 1998). Typical usage includes: Development of
algorithms, Data analysis, exploration and visualization, Mathematics and computing, Scientific and
technical graphics, Modeling, simulation and prototyping (Cavallo et al, 1996; Canos,2008;Bouchon
et al, 1995).
Fuzzy set theory deal with mathematically model information uncertainties and the theory has
been developed and applied in a number of real world applications (FuzzyTECK,1995; Chen, 2001;
Gebhardt, 1995; Zadeh,1965)
µ୅
: X ՜ ሾ0,1ሿ, x ՜ µ୅
ሺxሻ ‫א‬ ሾ0,1ሿ ሺ3ሻ
The value of the membership function µA(X) represents the membership grade of x in X. The
closer value to 1 is, the stronger degree of membership of x in A is. (Bector& Chandra 2005; Lu et
al.2007).
µA∪B(x)= max(µA(x), µB(x)) = µA(x) ∪µB(x) (4)
µA∩B(x)= min(µA(x), µB(x)) = µA(x) ∩ µB(x) (5)
µ Ac(x)=1- µA(x) (6)
2.2.2 FUZZY NUMBERS
A fuzzy number is one type of fuzzy sets with normalized membership function. A fuzzy
number ‫ܣ‬ሚ is subset of real line R who membership function µ஺෨(x) Can be a continuous mapping
from R into a closed interval [0,1]. The membership functionµ஺෨(x) has the following characteristics
(Dubois& Prade 1978;Wang et al. 2006).
The membership function of the number ‫ܣ‬ሚ can be expressed as follows.
µ஺෨(x) =
µ஺෩
௅ ሺ‫ݔ‬ሻ, ܽ ൑ ‫ݔ‬ ൑ ܾ
1, ܾ ൑ ‫ݔ‬ ൑ ܿ
µ஺෩
ோ ሺ‫ݔ‬ሻ, ܿ ൑ ‫ݔ‬ ൑ ݀
(7)
In a special case when b=c, the trapezoidal fuzzy number into a trianglar fuzzy number
(Abbasbandy & Hajjari 2009; Wang et al.2006).
ߤ஺෩
௅
ሺ‫ݔ‬ሻ= ೣషೌ
್షೌ
(8)
ߤ஺෩
ோ
ሺ‫ݔ‬ሻ ൌ ೏షೣ
೏ష೎
(9)
0,otherwise

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ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 12-20 © IAEME
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Figure.1: Trapezoidal and triangular fuzzy numbers
2.2.3 Fuzzy inference system FIS
The output level z of each rule is weighted by firing strength w of the rule (Guh et al, 2008;
Castillo, 1999a).The final output of the system is weighted average of all the rule output which is
given as:Final output=
∑ ௪೔ ௭೔
ಿ
೔సభ
∑ ௪೔
ಿ
೔సభ
(10)
The Sugeno proposed a fuzzy inference method that guarantees the continuity of the
output.Tomohiro Takagi and Michio Sugeno introduced a mathematical tool to build a fuzzy model
of a system where fuzzy implication and reasoning are used in their paper in the year of 1985
(Sugeno, 1985).A FISwith five functional blocks described in Figure.2.
Figure.2: Fuzzy inference system
3. CASE STUDY
The storage tank is designed to hold a flammable liquid under slight nitrogen positive
pressure under controls pressure (PICA-I)(CCPS,2000).The incident isthe top event that will be
developed in the fault tree, Construction of the fault tree FTbased on the knowledge of the system
andinitiating events in the hazard operability HAZOP study, the fault tree is constructed manually.
Equipment and valves Instruments
FV = Flow control Valve
T = Tank
P = pump
PV =Pressure control Valve
RV = Relief Valve
1῎
= 1 inch size
P = Pressure
T = Temperature
L = Level, F = Flow
I = Indicator
C = Controller
A = Alarm, H = High,
L = Low
µ஺෨(x)
db
0
ߤ஺෩
௅
ሺ‫ݔ‬ሻ ߤ஺෩
ோ
ሺ‫ݔ‬ሻ
1
x
ca 1
µ஺෨(x)
db =c
0
ߤ஺෩
௅
ሺ‫ݔ‬ሻ ߤ஺෩
ோ
ሺ‫ݔ‬ሻ
1
x
a 1
Database Rule base
Knowledge base
Fuzzification Defuzzification interface
Decision-making unit FuzzyFuzzy
Crisp
OUTPUTINPUT
Crisp

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Figure.3: Flammable liquid storage tank
Figure. 4 shows fault tree constructed manually. Every event is labeled sequentially with a T
for the top event, M for intermediate event, andB for a basic or undeveloped event. The procedure
starts at the top event, Tank rupture due to overpressure, and determines the possible events that
could lead to this incident (CCPS, 2000).
Figure.4: Fault tree analysis for tank rupture due to overpressure using boolean algebra
Tables.2 Shows calculate frequency using Boolean algebras and fuzzy inference methods, the
results by fuzzy sets are more precise than classical method , and in tables we can see two kinds of
results. The first illustrate using same imputs in classical with results more precise, the second is the
1
῎
PV-2 1
῎
PICA-1
H
PV-1
V-7
To atmosphere
Nitrogene
To flare
RV-1
V-8
1
῎
From tank
trucks
V-1
FlammableLiquid
Storage Tank
T=1
4
῎
LIA-1
TIA-1
V-3
1῎
P=1
PI-1
V-4
FICA
-1
H
H
L
FV-1
To process
L
Or
And
Exceed capacity of RV-1
B2=1.10
-3
/yr
V-8 closed
B3=1.10
-3
/yr
Tank rupture due to overpressure
T=2.10-5
/yr
Tank overpressured
M1=1.10-2
/yr
Pressure relief system failure
M2=2.10-3
/yr
Excess pressure in tank
M3=4.10
-5
/yr
Failure of PICA-I/
closingPV-1, B1=1.10
-
2
Hight pressure in tank
M4=4.10-3
/yr
Failure of, or ignoring
PICA-1, B4=1.10-2
/yr
Temperature of inlet hotter
than normal, B8=1.10-3
/yr
V-7 Closed
B7=1.10-3
/yr
High pressure in Flare
Headen, B6=1.10-3
/yr
PV-1 Fails closed
B5=1.10
-3
/yr
And
Or Or
1
2 3
4

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ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 12-20 © IAEME
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results by change the value of Input which influence directy in output. The results show the methodto
calculate top event as follow:
Table.2: Tank rupture due to overpressure using Boolean algebras and fuzzy inference methods
Calculate frequency using Boolean algebras method
If
Tank overpressured
=1.51*10-2
/yr And
Pressure relief system
failure = 2.5*10-3
Then
Tank rupture due to
overpressure=2*10-5
/yr
α-Level intervals of frequency usingfuzzy inference method
1
If
0.0151
And
0.0025
Then
2.53*10-5
2 [0.0101 0.0133] 0.0025 2.52*10-5
3 [0.0134 0.0201] 0.0025 2.53*10-5
4 0.0151 [0.002 0.003] 2.53*10-5
5 0.0101 [0.002 0.003] 2.52*10-5
Fault tree analysis after calculating the frequency using fuzzy inference method, the results
show that fuzzy sets is very appropriate for precise the value of initiating events for risk assessment.
Figure.5.a Rules inferencess process of Tank
rupture due to overpressure
Figure.5.bThree dimensional diagram, Tank
rupture due to overpressure , Pressure relief
system failure and Tank overpressured
Figure.5.c Two dimensional diagram, Tank
rupture due to overpressure with Tank
overpressured
Figure.5.dTwo dimensional diagram, Tank
rupture due to overpressure with Pressure
relief system failure

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ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 12-20 © IAEME
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Figure.6: Fault tree analysis for tank rupture due to overpressure using fuzzy inference
4. DISCUSSION
From our results above we can say that:
The results of fault tree analysis using fuzzy sets is more powerfull than classical methods,
which help us to precise the value of top event and to deal with uncertainty. The Surface Viewer
shows a three-dimensional curve that represents the mapping from inputs and outputs, the diagram
drawn using three equations essential (7)-(9).From the simulation results, it may be observed that the
fuzzy model can be successfully employed to instantaneously determine the exactitude of fault event
with only two input specifications.
The two-dimensional diagrams show the relationship between one input and one output,
whereas the three dimensional diagrams show the relationship between two inputs and one output
and the tables show also the change of outputs by changing the inputs.The two dimension diagram
for OR gate happened one steps as line continue, this show the extent of one input directly on output,
however the AND gate, the two dimensional diagramshow three steps which means that one input
not influence directly on output.The same analysis for three dimensional diagram for both OR and
AND gates.Results calculated of the rule output using equation (10).
5. CONCLUSION
In this paper, we have proposed a fuzzy fault tree analysis FFTA for reliability quantitative
risk assessmentas new model to solve problem of imprecise and uncertainty of results.The
application of this model gavebest results for the decision making.FFTA asnewmodelfor reliability
quantitative risk assessment based onFuzzy inference system FIS which considered as the best tool to
precise the value of top events of fault tree analysis.The fault tree analysis is constructed using two
different calculation approaches, boolean algebrasclassic and fuzzy inference using sugeno method,
Or
And
Exceed capacity of RV-1
B2=1.10
-3
V-8 closed
B3=1.10
-3
Tank rupture due to overpressure
T=2.53*10-5
/yr
Tank overpressured
M1=1.51*10-2
/yr
-5
Pressure relief system failure
M2=2.5410
-3
Excess pressure in tank
M3=4.54*10
-5
/yr
Failure of PICA-I/
closingPV-1, B1=1.10
-
2
/yr
Hight pressure in tank
M4=4.10
-3
/yr
Failure of, or ignoring
PICA-1, B4=1.10
-2
/yr
Temperature of inlet hotter
than normal, B8=1.10-3
/yr
V-7 Closed
B7=1.10
-3
/yr
High pressure in Flare
Headen, B6=1.10
-3
/yr
PV-1 Fails closed
B5=1.10-3
/yr
And
Or Or
1
2 3
4

8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print),
ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 12-20 © IAEME
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whereas the fuzzy inference system gave results more precise than boolean algebras method, which
consider as comlementry to solve problems of uncertainty.Fault tree analysis with fuzzy sets is good
solution for reliability quantitative risk assessment.
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