Quantitative Risk Analysis Model of Integrating Fuzzy Fault Tree with Bayesian

Network

Van Fu WANG, Min XIE, Kien MingNG

Department of [ndustrial & Systems Engineering,

National University of Singapore,

10 Kent Ridge Crescent, Singapore

Abstract - In this paper, a new quantitative risk analysis

model of integrating fuzzy fault tree (FFT) with Bayesian

Network (BN) is proposed. The first step involves describing

a fuzzy fault tree analysis technique based on the Takagi and

Sugeno model. The second step proposes the translation

rules for converting FFT into BN. Based on this, the

integration algorithm is demonstrated by an offshore fire

case study. The example clearly shows that FFT can be

directly converted into BN and the classical parameters of

FFT can be obtained by the basic inference techniques of

BN. By using the advantages of both techniques, the model

of integrating FFT with BN is more flexible and useful than

traditional fault tree model. This new model not only can be

used for describing the causal effect of accident escalation

but also for computing the occurrence probability of

accident based on historical data and fuzzy logic.

Keywords - Quantitative Risk Analysis Model, Fuzzy

Fault Tree, Bayesian Network

I. [NTRODUCTION

In conventional fault tree analysis (FTA), the probabilities of the basic events are treated as exact values. However, in many cases, it is generally difficult to

estimate the precise probabilities of basic events from the

past experiences. So, it is necessary to develop a new method to capture the imprecision of failure data for use

in the FT A. [t may be more appropriate to use fuzzy numbers instead of the exact probability values [1] Extensive research has been conducted through applying fuzzy sets theory in FTA. Reference [2] is the earliest to

research fuzzy fault tree (FFT), which treated

probabilities of basic events as trapezoidal fuzzy numbers, and applied the fuzzy extension principle to determine the

occurrence probability of TE. Reference [3] analyzed fuzzy reliability using L-R type fuzzy numbers. [n order

to facilitate the calculation of Singer's method, revised

methods to analyze the fault tree (FT) by specifically

considering the failure probabilities of basic events as triangular fuzzy numbers is proposed [4]. Reference [5] adopted possibility theory to analyze FFT. Intuitionistic

fuzzy methods are used to analyze FT on printed circuit board assembly [6]. Reference [7] presented a method that overcomes the drawbacks of traditional FTA through

fuzzy fault tree analysis (FFTA) based on possibility measures and fuzzy logic. A novel FT A technique based

on the Takagi and Sugeno (T -S) model is proposed [8]. Since the proposed TS-FT A is derived from fuzzy logic

This paper is sponsored by Singapore's Agency for Science, Technology and Research (A*STAR) and Poland's Ministry of Science and Higher Education(MSHE) (SERC grant number 0721340050)

978-1-61284-4577-0085-9111/$26.00 ©2011 IEEE

Yi Fei MENG

Department of Chemical Engineering,

China University of Petroleum,

Qingdao 266555, PR China

and the T -S model, it can readily handle fuzzy

information and uncertainties in the relationships among

events. It is obvious from the above reviews that FFT A has

been applied to many real problems effectively. FFTA can offer an efficient method of representing the fault causes

and cope with the existence of several faults at the same time. However, FFTA does not have the same capability

of incorporating the evidence into the reasoning that Bayesian Network (BN) has. Moreover, FFTA is entirely

deterministic and no evidence can be given without re­evaluating the FT [9].

BN model is a tool used to manage uncertainty using

probability [10]. A fundamental assumption for BN is that

the strength of the interaction/influence among the graph

nodes is uncertain, and thus this uncertainty is represented by assigning a probability of existence to each of the links

joining the different nodes[ll].Updating the probability is

possible when observation is performed on the system

[12]. Reference [13] proposed the conversion thinking

from FT to BN. This conversion algorithm is innovative but their probability calculation of FT and BN are insufficient. Reference [14] extended FTs by defining the

time-to-failure of FT as deterministic functions of the

corresponding input components' time-to-failure.

BN can provide a relatively simple and updatable fire

dynamic model that can accommodate uncertainty of the

relationships among basic events [15, 23]. However, BN is unable to determine accurately how the failures cause the undesired fault, which is the advantage of FT [9]. Thus, in this paper, we attempt to extend the use of FFT to be integrated with BN, for which the probability

calculation is more flexible and convenient.

[I. METHODOLOGY

A. Fuzzy Fault Tree Analysis

Because the exact causes of accident may not always be known in many cases, the conventional gates of FT are

not suitable when there are uncertainties in the

relationships among events. To handle these uncertainties, a new gate based on the Takagi and Sugeno (T -S) model

is presented. The failure possibility of the TE can be computed by the T -S model from the fuzzy possibilities of the basic events, discussed as follows.

T-S model include a set of IF-THEN fuzzy rules, which

can be used to describe the relationships among events.

267

So it can be used to construct the T -S Gate. Consider the I th rule of the T -S model [16]: Suppose the magnitude of

the failure possibility of the basic events Xv x2, ... , Xn and super events Y are denoted respectively by: (xi, xL ... , xt1), (x�, x�, ... , X�2), ... , (x�, x�, ... , x�n) and

yl, y2,... , yky , which satisfy the following equation:

o ::; xi < xi < ... < xt 1 ::; 1 o < xi < x� < ... < X�2 ::; 1

o ::; xA < x� < ... < x�n ::; 1 o ::; y1 < y2 < ... < yky::; 1

Then the T -S gate can be represented by the following fuzzy rules, Rule I (I = I, 2, ... ,m) :

If Xl is xiI, and X2 is X�2, . . . , and Xn is xhn, then the possibility of y1 is pl(y1) , y2 is pl(y2) , ... , yky is pi (ykY) , where i1 = 1,2, ... , k1; i2 = 1,2, ... , k2; in =

1,2, ... ,kn; m is the number of fuzzy rules, m =

k 1k2 ... kn· In order to obtain the fuzzy rules, the possibility

magnitude of basic events should be defined according to the historical data and/or experts' experience. And then the fuzzy failure possibilities of TE can be obtained from the basic events using fuzzy logic.

Suppose the possibility magnitude of basic events is x' = (x�, x;, ... , x�), the fuzzy failure possibilities of the TEs can be obtained by the T -S gate [8],

where,

ill

p(y1) = L>� (x')pl(y1) 1= 1

ill

p(y2) = I p�(x')pl(y2) 1= 1

ill

p(yn) = I p�(x')pl(yn) 1= 1

(:1.* ( ') =

rr)=l [lex)) t-'I X " ill rrn (') LI=l j=l [l Xj

and [lex)) is the membership of xj for the corresponding fuzzy set.

Suppose the fuzzy possibility of the basic events is: p(xi 1 )(i1 = 1,2, ... ,k 1) p(x�2)(i2 = 1,2, ... ,k2) ... p(xhn)(in = 1,2, ... , kn), then the possibility of the rule I is : pi = p(xi1) p(X�2) ... p(xhn) (I = 1,2 ... m) .

The fuzzy possibility of the TE is calculated by:

m

p(y 1) = I pl. pl(y1) 1= 1

m

p(y2) = I pl. pl(y2) 1= 1

m

p(yn) = I pl. pl(yn) 1= 1

The "AND" and "OR" gates in the conventional FT A can be implemented by the T -S gate. The "AND" gate can be represented by the following fuzzy rule: If Xl is 1, and X2 is 1, and . . . , and Xn is 1, then the possibility ofy1

= 0 is 0, y2 = 0 is 0, ... , the possibility ofyky = 1 is 1.

The "OR" gate can be represented by the following rules: { Rule I (I = 1, 2, ... , m) :

If XI is 1 , then the possibility of y1

= 0 is 0, y2 = 0 is 0 , ... , the possibility of yky = 1 is 1.

B. Bayesian Network

BN is a probabilistic graphical model that represents a set of random variables and their relations via a directed acyclic graph (DAG) [17]. BNs are known to be useful in assessing the probabilistic relationships and identifying probabilistic mappings between system events [18]. The graphical structure of a BN depicts a qualitative illustration of the interactions among the random variables. Numerically, a BN represents the joint probability distribution among the modeled variables [19]. The variables of a BN are defined as the causes of accidents, while the links represent the interaction of the failure events. The conditional independence assertions about the variables, represented by the absence of arcs, allow decomposition of the underlying joint probability distribution. This significantly reduces the complexity of inference tasks on the BN [14].

Let G = (V, E) be a DAG and X = (Xv) v L V be a set of random variables indexed by V. X is a BN with respect to G if its joint probability density function can be written as a product of the individual density functions, conditional on their parent variables [20]:

P(x) = n P(xvlxpa(v)) VEV

where, pa (v) is the set of parents of v. For any set of random variables, the probability of any member of a joint distribution can be calculated from conditional probabilities using the chain rule as follows:

P(X1 = xv .. · , Xn = xn) = rr�=l P(Xv = Xv/XV+1 =

xv+v ... ,Xn = xn)

268

From the joint probability distribution P(X1 =x1, ... ,

Xn=xn), various marginal and conditional probabilities

can be computed [21].

C. Translation rules for converting fuzzy fault tree to Bayesian network

Generally, traditional FT can be converted into a BN,

and that inference technique of BN may be used to obtain classical parameters of FT [13]. The dynamic FT can also

be mapped into an equivalent discrete-time Bayesian

network [24]. Similarly, FFT can also be transformed into

Bayesian network for convenient analysis. Specifically, a FFT can be transformed into an

equivalent BN as follows: • Each basic event of the FFT is converted into a

corresponding child node in the BN. The child nodes are triple (high/medium/low) and let the failure

magnitude of corresponding state be 1,0.5 and °

respectively (arbitrary values stand for different states );

• Each child node in the BN is assigned the same prior

probability as the possibility magnitude of the corresponding basic event in the FFT;

• For each gate of the FFT, a relevant parent node in

the BN is created;

• Child nodes in the BN are connected to the parent

node as the basic events are connected to the

associated gates in the FFT; • Each parent node of the BN is assigned to the same

conditional probability table (CPT) as the relevant fuzzy T -S gate in the FFT. The Boolean functions

modeled by the gates in the FFT are modeled by

parent nodes in the BN;

X6

ignition 1M 1

X7 X8

X5

Jet fire ITO

@

><4

• The parent nodes in the BN are connected as the relevant gates are connected in the FFT.

III. CASE STUDY

A. Fuzzy fault tree analysis In this section, the integration algorithm mentioned

above is demonstrated by an offshore fire case study. Firstly, the most credible fire scenario for compressor unit of offshore production is analyzed using FFT A:

flammable gas continuously release from compressor or

its pipeline of offshore platform and it would cause a jet

fire if the released gas is ignited along with releasing. The

jet fire may cause other units to fail [22]. Secondly, the

FFT is converted to BN and the BN model is analyzed using software. Finally, the occurrence probability of jet

fire scenario and the posterior probabilities of all basic events are predicted by BN inference.

FFT is constructed to analyze causal effect of the jet fire in Fig.l.

The failure magnitude of the basic events is defined according to historical data and experts' experience, e.g.,

High, Medium and Low correspond to 1, 0.5 and °

respectively. The possibilities of basic events and yearly

failure frequency of each basic event, where the magnitude of possibility of basic events is 1, are shown in Table I.

Those frequency data of basic events were collected

from Offshore Reliability Data Handbook and World­

wide Offshore Accident Databases. Suppose the fuzzy possibilities of all basic events failing with a magnitude of

0.5 are the same as that with a magnitude of 1. Taking M1 as example, how the IF-THEN fuzzy rules can

constitute a T-S gate is demonstrated in Table II.

X2 X12

Xl ><9 Xl0 X"

Figure I. Fuzzy fault tree of Jet fire

TABLE!. X5 Release from upstream pipeline 0.024 0.952

BASIC EVENTS OF FAULT TREE X6 Ignition due to explosion energy 0.075 0.85 X7 Ignition due to surrounding heat 0.1 0.8

Basic events possibility possibility X8 Ignition due to electric spark 0.125 0.75 ofX=1 ofX=O X9 Release junction of pump and pipeline 0.005 0.99

XO Release from downstream pipeline 0.003 0.994 XIO Release from rotor 0.03 0.94

XI Compressor failed causing release 0.035 0.93 XII Pump failed to operate 0.075 0.85

X2 Release from impeller 0.05 0.9 XI2 Release from casing 0.1 0.8

X3 Release from seal 0.06 0.88 X4 Release from casing of compressor 0.025 0.95

269

TABLE II.

T-SGATEMI

P(M1 = 0) = Lf�l pl. pl(Ml = 0) = Lf�l pi (X6) . pl(X7)' pl(X8)· pl(Ml = 0) = 0.6183.

MI The fuzzy possibilities of other T -S gates are

----:-R_ul_e_

--::X_6 __

----::X_7 ___ X:-8 __ ----,.I-,-::-__

--:O,..,

.5:------::

0:-:-::

__ calculated using the same method, The calculation results

I 0 0 I 0.65 0.2 0.15 are shown in Table III. 2 0 0 0.5 0.25 0.5 0.25 3 0 0 0 0 0 1 4 0 0.5 I 0.85 0.1 0.05 5 0 0.5 0.5 0.55 OJ 0.15 6 0 0.5 0 0.l5 0.4 0.45 7 0 I I 0.9 0 0.1 8 0 I 0.5 0.8 0.1 0.1 9 0 1 0 0.7 0.2 0.1 10 0.5 0 I 0.8 0.1 0.1 II 0.5 0 0.5 0.6 0.4 0 12 0.5 0 0 0.2 0.5 OJ 13 0.5 0.5 I I 0 0 14 0.5 0.5 0.5 0.7 OJ 0 15 0.5 0.5 0 0.6 0.4 0 16 0.5 I I 0.95 0 0.05 17 0.5 I 0.5 0.8 0.1 0.1 18 0.5 1 0 0.7 0.3 0 19 I I I I 0 0 20 I I 0.5 I 0 0 21 1 1 0 1 0 0 22 I 0.5 I I 0 0 23 I 0.5 0.5 I 0 0 24 1 0.5 0 1 0 0 25 I 0 I I 0 0 26 I 0 0.5 I 0 0 27 1 0 0 1 0 0

The fuzzy possibility of Ml can be obtained according

to T -S model as follows:

P(M1 = 1) = Lt�l pl. pl(Ml = 1) = Lt�l pl(X6) . PIX7, PIX8'PIMl=1=O,2079;

P(M1 = 0.5) = Lf�l pl. pl(Ml = 0,5) = Lf�l pl(X6) . pl(X7)' pl(X8) . pl(Ml = 0.5) = 0,1738;

TABLE III.

FUZZY POSSIBILITY OF EACH T -S GATE

T -s gate High(l) Ml 0.2079 M2 0.0693 M3 0.0572 M4 0.094 M5 0.0168 TO 0.0755

Medium(0.5) 0.l738 0.0837 0.0362 0.0867 0.179

0.0951

Low(O) 0.6183 0.847

0.9066 0.8193 0.8042 0.8294

B. Converting fuzzy fault trees to Bayesian networks

The BN corresponding to the above FFT is depicted in

Fig.2. The CPTs of BN are the same as the above T-S gate of FFT, The CPT of node "ignition" is the same as the data of Ml gate, which is shown in Table II. The

CPTs of other nodes are calculated using the same method, After all CPTs are elicited, we can perform the probability analysis using Bayesian inference,

From the Bayesian inference, we can know that the fuzzy possibility of TE is:

P(Io = 1) = 0.0824 P(Io = 0,5) = 0.0998

P(Io = 0) = 0.818 The conditional probabilities of basic events are presented

in Table IV,

Figure 2. Bayesian network ofJet fire

TABLEIY. X4 0,025 3J7 0.0002898 CONDITIONAL PROBABILITY OF BASIC EVENTS X5 0.024 3.29 0.0002918

X6 0.075 0 0.02431 Prior Probability(X Posterior Entropy reduction X7 0,1 4.17 0.01316 at high state) Probability (%) X8 0,125 43 0.0822

XO 0,003 0.41 3.58ge-005 X9 0.005 0.62 2.944e-005 Xl 0.035 4.71 0.0004035 XIO 0,03 3.72 0.0001785 X2 0,05 7.17 0.0008902 XII 0,075 9.68 0.0006508 X3 0,06 8.09 0.0007129 X12 0.1 12.4 0.0006254

270

From Table IV, it can be seen that event X5, X6 and

X7 give more contribution to the occurrence of the eventual accident. This analysis shows that particular attention should be paid to "explosion energy",

"surrounding heat" and "electric spark", as these events

are most likely to cause eventual jet fire.

IV. CONCLUSION

In this paper, we presented an approach of making FFT

to be logically and probabilistic ally integrated with BN.

The case study clearly shows that FFT can be directly

converted into BN and the classical parameters of FFT

can be obtained by the basic inference techniques ofBN. The use of fuzzy logic in FFT A allows fuzzy

information such as linguistic information to be incorporated. It can deal with uncertainties of the failure causes, due to insufficient knowledge of the relationships

among basic events. Therefore the FFTA is more suitable

for analyzing the failure causes than the traditional FT A. BN is useful for assessing the probabilistic relationships

and identifying probabilistic mappings between basic events. It can also be used for predicting the marginal

posterior probabilities of each basic event, which are often used to identify the criticality of basic events. Controlling the occurrences probability of these crucial events would

considerably reduce the probability of eventual accidents. The model of combining FFT and BN is more flexible

and useful than traditional FT model. This novel model

not only can be used for describing the causal effect of accident escalation but also for computing the probability of accident based on historical data and fuzzy logic.

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