The weakest t-norm based intuitionistic fuzzy fault-tree analysis to evaluatesystem reliabilityMohit Kumar ∗, Shiv Prasad YadavDepartment of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
a r t i c l e i n f o
Article history:Received 23 October 2011Received in revised form14 January 2012Accepted 15 January 2012Available online 22 March 2012
In this paper, a new approach of intuitionistic fuzzy fault-tree analysis is proposed to evaluate systemreliability and to find themost critical system component that affects the system reliability. Here weakestt-norm based intuitionistic fuzzy fault tree analysis is presented to calculate fault interval of systemcomponents from integrating expert’s knowledge and experience in terms of providing the possibility offailure of bottom events. It applies fault-tree analysis, α-cut of intuitionistic fuzzy set and Tω (the weakestt-norm) based arithmetic operations on triangular intuitionistic fuzzy sets to obtain fault interval andreliability interval of the system. This paper also modifies Tanaka et al.’s fuzzy fault-tree definition. Innumerical verification, a malfunction of weapon system ‘‘automatic gun’’ is presented as a numericalexample. The result of the proposed method is compared with the listing approaches of reliabilityanalysis methods.
The high tech industry changes very fast and grows morecomplicated day by day. To satisfy the demand and increasethe quality of life for human beings, the high tech industry isdevoted to improve its own quality to fit the expectations ofconsumer and upgrade competitiveness of the products on themarket. Therefore, to any one system, improving the quality andthe competitiveness of the products, the product reliability is animportant issue both on the academic research and practice. Inreal world, weapon systems are one of the most complicatedsystem products. In order to integrate sophisticated functionsunder system command and control, weapon systems includemany different system components. Weapon system reliabilityproblem is critical and important, because it is not only expensiveproduct but also it might change history of a war or combat due toits ability at a specific time and in space.
Fault tree analysis (FTA) is a powerful diagnosis technique andis used widely for finding the most critical system componentin system failure, and logical functional relationship amongcomponents, manufacturing processes, and subsystems of asystem [1–3]. In the objective world have two kinds of uncertaintyfactor: randomuncertainty and fuzzy uncertainty. Traditional faulttree analysis based on Boolean algebra and probability theory, has
solved the randomuncertainty problem verywell. But in the large-scale the complicated system has the massive fuzzy uncertainty, itis necessary to get the exact probability of the basic case is difficult.Therefore, the introduction of fuzzy theory and technologynot onlyhas important theoretical significance, but also the urgent need forpractical engineering.
Tanaka et al. [4] used trapezoidal fuzzy number to replaceprobability and applied fault-tree analysis to obtain the system’sfault interval. Singer [5] presented a fuzzy set approach torepresent the relative frequencies of the basic events. Hedemonstrates the use of the n-array possibilistic AND, OR andNEG operators to construct possible fault-trees. Lee et al. [6]proposed a fuzzy fault-tree analysis method to solve newequipment’s maintenance problem. Cai [7] gave a differentinsight by introducing the possibility assumption and fuzzystate assumption to replace the probability and binary stateassumptions. The concept of the importance used in the fuzzy faulttree was introduced by Tsujimura and Gen [8], where the failurepossibilities of basic events were considered as fuzzy numbers.Fuzzy possibility was proposed by Walley [9] and then there weresome approaches about fuzzy fault-tree analysis being introducedin [10,11].
Out of several higher-order fuzzy sets, intuitionistic fuzzy set(IFS) first introduced by Atanassov [12] has been found to becompatible to deal with vagueness and uncertainty. Due to thesedistinct features in characterizing vagueness and uncertainty, itis natural to expect that IFSs play a significant role in enrichingreliability analyzing. The concepts of IFS can be viewed as anappropriate/alternative approach to define a fuzzy set in the casewhere available information is not sufficient for the definition
532 M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538
of an imprecise concept by means of a conventional fuzzyset. In fuzzy sets the degree of acceptance is considered onlybut IFS is characterized by a membership function and a non-membership function so that the sum of both values is lessthan or equal to one. In fact, Biswas [13] pointed out that therewere situations where IFS theory is more appropriate to deal.Bustince and Burillo [14] proposed that the concept of vague setscoincides with that of intuitionistic fuzzy sets (IFSs). Therefore,it is expected that IFSs ould be used to simulate any activitiesand processes requiring human expertise and knowledge, whichare inevitably imprecise or not totally reliable. IFS theory hasbeen applied in different areas such as logic programming [15,16], decision-making problems [17,18], in medical diagnosis [19],and pattern recognitions [20]. Chen [21] presented a method foranalyzing the fuzzy system reliability based on triangular vaguesets. Shu et al. [22] proposed a method for the failure analysisproblem of printed circuit board assembly (PCBA) to computethe intuitionistic fuzzy fault-tree interval, traditional reliability,and the intuitionistic fuzzy reliability interval. Chang et al. [23]proposed a vague fault-tree analysis procedure to determine theweapon system’s reliability. Recently, Cheng et al. [24] proposedan intuitionistic fault-tree analysis procedure to determine theintuitionistic fuzzy reliability interval for liquefied natural gasterminal emergency shutdown system. Their approach integratedexperts’ knowledge and experience in terms of providing thepossibility of failure of bottom events, and used a triangular IFS toperform the calculation.
This paper proposes a new approach to determine theintuitionistic fuzzy reliability interval for systems using Tω (theweakest t-norm) based arithmetic operations on triangular IFSs.Also it modifies Tanaka et al.’s fuzzy fault-tree definition [4].Comparing the results of existing approaches [2,4,24,25] and theproposed approach, it has been shown that length of reliabilityinterval (uncertainty about the reliability) decreases using theproposed approach and obtained results are exact. While using theexisting approaches results are approximate due to approximateproduct of triangular IFSs.
The rest part of this paper is organized as follows. In Section 2,we give the review of basic concepts related to intuitionistic fuzzysets and its operations. Then intuitionistic fuzzy fault diagnosismodels are proposed. The proposed approach for intuitionisticfuzzy fault tree analysis is represented in Section 3. In Section 4,weapon system ‘‘automatic gun’’ is used to illustrate the proposedapproach of intuitionistic fuzzy fault-tree analysis and somecomparative studies are conducted, followed by some concludingremarks in Section 5.
2. Intuitionistic fuzzy set and its operations
In this section, we introduce the definitions and properties ofIFS, and four Tω (the weakest t-norm) based arithmetic operationson triangular IFSs.
2.1. Intuitionistic fuzzy set theory
Fuzzy set theory was first introduced by Zadeh [26] in 1965.A fuzzy set defined on the universe of discourse U,U = {u1,u2, . . . , un}, is a set of ordered pairs {(u1, µA(u1)), (u2, µA(u2)),. . . , (un, µA(un))}, where µA is the membership function of theclassical fuzzy set A, µA : U → [0, 1], and µA(uj) is the grade ofmembership of uj in A, ∀ uj ∈ U . The membership value µA(uj) isa real value between 0 and 1. It indicates the evidence for uj ∈ U ,but does not indicate the evidence against uj ∈ U . Fuzzy set theoryhas been shown to be a useful tool to handle such situations byattributing a degree to which a certain object belongs to a set. Inreal life, a person may assume that an object belongs to a set to
Fig. 1. IFS explanation of real number R.
Fig. 2. Triangular IFS.
a certain degree, but it is possible that he is not so sure about it.In other words, there may be a hesitation or uncertainty aboutthe membership degree of x in A. In fuzzy set theory, there is nomeans to incorporate that hesitation in the membership degrees.To incorporate that hesitation in the membership degrees IFS isused. Atanassov [27,28] introduced the concept of an IFS as ageneralization of ordinary fuzzy set.
Definition 2.1.1. Let X be a universe of discourse. Then anIFS Ai in X is a set of ordered triples given by Ai
=⟨x, µAi(x), νAi(x)⟩ : x ∈ X
, whereµAi : X → [0, 1] and νAi : X →
[0, 1] are functions such that 0 ≤ µAi(x)+ νAi(x) ≤ 1, ∀x ∈ X . Foreach x ∈ X , the numbers µAi(x) and νAi(x) represent respectivelythe degree of membership and degree of non-membership of theelement x ∈ X to Ai
⊆ X . For each x ∈ X , the value πAi(x) = 1 −
µAi(x) − νAi(x) is called the intuitionistic fuzzy index (uncertaintyor hesitation) of x in Ai. It represents the degree of non-determinacyor uncertainty of x ∈ X to IFS Ai. If the πAi(x) is little, it representwe are more certainly about x. If πAi(x) is great, it represent we aremore uncertainly about x. Obviously, when µAi(x) = 1 − νAi(x)for all elements of the universe, the traditional fuzzy set concept isrecovered. Fig. 1 shows IFS of real number R [29].
Definition 2.1.2. A triangular IFS of the universe of discourse X ,as shown in Fig. 2, is denoted by Ai
=
m, α, β, α′, β ′T ; µ; ν
where 0 ≤ µ ≤ ν ≤ 1 and defined bymembership functionµAi(x)
M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538 533
and non-membership function νAi(x) as
µAi(x) =
µ
1 −
m − xα
ifm − α ≤ x ≤ m, ∀x ∈ X
µ
1 −
x − mβ
ifm ≤ x ≤ m + β, ∀x ∈ X
0 otherwise
νAi(x) =
1 − ν
1 −
m − xα′
ifm − α′
≤ x ≤ m, ∀x ∈ X
1 − ν
1 −
x − mβ ′
ifm ≤ x ≤ m + β ′, ∀x ∈ X
1 otherwise
where m ∈ X is the center, α > 0 and β > 0 are called leftand right spreads of membership function of Ai respectively. Alsoα′ > 0 and β ′ > 0 represent left and right spreads of non-membership function of Ai respectively.
2.2. Tω-based arithmetic operations on triangular intuitionistic fuzzysets
In this section, we define Tω (the weakest t-norm) basedarithmetic operations on triangular IFSs [30]. Let Ai and Bi are twoIFSs and T = Tω be the weakest t-norm.
Ai=
m1, α1, β1, α
′
1, β′
1
T ; µAi; νAi
(1)
Bi=
m2, α2, β2, α
′
2, β′
2
T ; µBi; νBi
. (2)
Let µAi = µBi and νAi = νBi , then arithmetic operations on Ai
Tanaka et al.’s [4] considered fuzzy set defined on [0, 1] inFTA. In this paper, triangular IFS is considered for analyzing a faulttree. In terms of implementing Tω-based arithmetic operations ontriangular IFSs in FTA, this paper modifies Tanaka et al.’s [4] fuzzyFTA definition and re-defines influence degrees of every bottomevent (i.e. the leaf node in the fault tree) as the following:
Definition 3.1. Let qT denote the possible failure interval of the topevent and qTj represents that qT does not include the jth basic eventof a failure interval (delete the jth unit). V denotes the differencebetween qT and qTj . The larger value of V represents that jth basicevent has a greater influence on qT ; then
V (qT , qTj)
=
mT − α′
T
−
mTj − α′
Tj
+
(mT − αT ) −
mTj − αTj
+
mT − mTj
+
(mT + βT ) −
mTj + βTj
+
mT + β ′
T
−
mTj + β ′
Tj
(6)
V (qT , qTj)
= 5mT − mTj
+
αTj − αT
+
βT − βTj
+
α′
Tj − α′
T
+
β ′
T − β ′
Tj
where qT =
mT , αT , βT , α
′
T , β′
T
T and qTj =
mTj , αTj , βTj , α
′
Tj,
β ′
Tj
T .
In order to implement intuitionistic fuzzy fault-tree analysis inweapon system, this paper proposes five steps. The following stepsare also the basis of model for constructing intuitionistic fuzzyfault-tree analysis.
Step 1. Construct fault-tree logic diagram.To construct fault-tree diagram by fault-tree logicalsymbols and tracing backwhole process from top to bottomevents (as Fig. 3).
Step 2. Obtain possible failure interval of bottom event.Possible failure intervals of bottom events are obtained byaggregating experts knowledge and experience.
Step 3. Calculate possible failure interval of systems using Tω-basedarithmetic operations on triangular IFSs.From fault-tree diagram and possible failure intervals ofbottom events, this step can calculate possible failureinterval of systems using Tω-based arithmetic operationson triangular IFSs as Eqs. (3)–(5) to obtain the failureinterval of top event.
Step 4. Compute the reliability interval of top event.The reliability interval of top event is equal to one minusthe failure interval of top event.
Step 5. Find the most influential bottom event of system reliability.By the definition in this section, we delete the ith bottomevent in the fault-tree diagram, and calculate V (qT , qTj), ∀jto find the most influential power (i.e. maxj V (qT , qTj)) forthe whole system. (The results are shown as Table 2 inSection 4.5.)
Step 6. Analyze the results and suggestions.
4. Numerical verification
In this section, weapon system ‘‘automatic gun’’ [23] is takenfor numerical verification of proposed approach. In order toimplement arithmetic operation, a case study of automatic gunis presented. To construct fault tree including the top event(‘‘automatic gun cannot fire’’), the second events (‘‘firing assemblyfailure’’, ‘‘manual mistakes’’, and ‘‘feeder block’’), and the bottomevents (‘‘manual mistakes’’, ‘‘body failure’’, ‘‘extractor failure’’,‘‘spring failure’’, ‘‘feed frame failure’’, ‘‘out of machine oil’’, ‘‘pinfiring too short’’, ‘‘bad weather’’, ‘‘machine oil filter dirty’’, ‘‘airfilter dirty’’, ‘‘drive distortion’’, ‘‘copper dirt jam’’, and ‘‘oil dirtjam’’), fault tree integrates the top event, the second event, and thebottom event with ‘‘OR’’ and ‘‘AND’’ gate (Fig. 3). After fault treeis constructed, the possibility failure interval of bottom events isgenerated in Table 1 from expert’s knowledge and experience.
For connecting the fault tree diagram of ‘‘automatic gun cannotfire’’, this research uses logical node to describe ‘‘AND’’ gate withthe sign of ∩, and ‘‘OR’’ gate with the sign of ∪. It can representtheir relationship of parallel and series.
T = R ∪ A ∪ S= (B ∪ W ∪ X) ∪ A ∪ (C ∪ D ∪ E)
= (B ∪ (F ∪ Y ) ∪ (G ∪ Z)) ∪ A ∪ (C ∪ D ∪ E)
= (B ∪ (F ∪ (H ∩ I ∩ J)) ∪ (G ∪ (K ∩ L ∩ M)))
∪ A ∪ (C ∪ D ∪ E)
where ∪ means relation of parallel and ∩ means relation of series.Let qj represent the failure possibility of bottom event j, then
the failure possibility of R is:
qR = 1 − (1 − qB) (1 − qW ) (1 − qX ) .
The failure possibility of S is:
qS = 1 − (1 − qC ) (1 − qD) (1 − qE) .
The failure possibility ofW is:
qW = 1 − (1 − qF ) (1 − qY ) .
The failure possibility of X is:
qX = 1 − (1 − qG) (1 − qZ ) .
534 M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538
3)
4)
5)
Ai⊕T Bi
=
m1 + m2,max (α1, α2) ,max (β1, β2) ,max
α′
1, α′
2
,max
β ′
1, β′
2
T ;
minµAi , µBi
,max
νAi , νBi
(
Ai⊖T Bi
=
m1 − m2,max (α1, β2) ,max (β1, α2) ,max
α′
1, β′
2
,max
β ′
1, α′
2
T ;
minµAi , µBi
,max
νAi , νBi
(
Ai⊗T Bi
=
m1m2,max (α1m2, α2m1) ,max (β1m2, β2m1) ,max
α′
1m2, α′
2m1,max
β ′
1m2, β′
2m1
T ;
minµAi , µBi
,max
νAi , νBi
for m1,m2 > 0
m1m2,max (β1m2, β2m1) ,max (α1m2, α2m1) ,maxβ ′
1m2, β′
2m1,max
α′
1m2, α′
2m1
T ;
minµAi , µBi
,max
νAi , νBi
for m1,m2 < 0
m1m2,max (α1m2, −β2m1) ,max (β1m2, −α2m1) ,maxα′
1m2, −β ′
2m1,max
β ′
1m2, −α′
2m1
T ;
minµAi , µBi
,max
νAi , νBi
for m1 < 0,m2 > 0
0, α1m2, β1m2, α′
1m2, β′
1m2T ;min
µAi , µBi
,max
νAi , νBi
for m1 = 0,m2 > 0
0, −β1m2, −α1m2, −β ′
1m2, −α′
1m2T ;min
µAi , µBi
,max
νAi , νBi
for m1 = 0,m2 < 0
(0, 0, 0, 0, 0)T ;minµAi , µBi
,max
νAi , νBi
for m1 = 0,m2 = 0.
(
Box I.
Table 1The possible range of bottom event failure.
Traditionally, probability method [2] is the method for dealingwith the heterogeneous problems, and probability can only showthe randomness of success or failure events. This method isconstrained to its usage on the condition of great amount of datasample and all of event outcomes are under certainty. However,a lot of uncertainty factors cause fuzziness in the procedure ofweapon system evaluation, for example: statistics uncertainty,model uncertainty, and data uncertainty. These uncertainty factorswill limit the understanding of system component failure dueto the reason of incomplete data. Also, the traditional reliabilitymethod is lack of ability to make statistical estimate. Therefore,traditional reliability method is hard to calculate failure possibilityof system and its component in a precise way because of theincomplete data. This research calculated the failure possibility oftop event ‘‘automatic gun cannot fire’’ based on data of Table 1(columnm) as follows.
M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538 535
Fig. 3. Intuitionistic fuzzy fault tree.
After the above calculation, we find that the failure probabilityof top event ‘‘automatic gun cannot fire’’ is 0.037396 and thereliability of ‘‘automatic gun’’ is 0.962604.
4.2. Huang et al.’s method
When the failure probability of a system is extremely smallor when essential statistical data are scarce, the posbist fault-tree analysis proposed by Huang et al. [25] could be applied topredict and diagnose a system’s failures and evaluate its reliabilityand safety. Calculations of the failure possibility of top event‘‘automatic gun cannot fire’’ based on the crisp failure possibilitieslisted in Table 1 (columnm), per the following:
Poss(Y ) = min (Poss(H), Poss(I), Poss(J))= min (0.009, 0.005, 0.007)= 0.005
Poss(Z) = min (Poss(K), Poss(L), Poss(M))
= min (0.004, 0.009, 0.002)
= 0.002.
Then, the top event failure possibility can be calculated as
Poss(T ) = max (Poss(A), Poss(B), Poss(C), Poss(D), Poss(E),
Poss(F), Poss(G), Poss(Y ), Poss(Z))
= max (0.005, 0.006, 0.005, 0.008, 0.007,0.004, 0.003, 0.005, 0.002)
= 0.008.
After the above calculation, it is shown that the failurepossibility of the top event ‘‘automatic gun cannot fire’’ is 0.008and the reliability of the ‘‘automatic gun’’ is 0.992.
4.3. Tanaka et al.’s method [4]
It is often difficult to assign a unique numerical value between0 and 1 to a failure probability. To circumvent this difficulty,the failure probability can be defined as a fuzzy set on [0, 1].Specifically, the possibility of failure defined in a certain range on
536 M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538
[0, 1] is used instead of a unique value of probability. Now theproblem is to calculate the failure possibility of the top event asa fuzzy set, given the failure possibilities of fundamental events.Instead of a specific value of probability, Tanaka et al. dealt witha fuzzy number on [0, 1], viz. fuzzy-probability or possibility offailure.
This procedure calculates the failure possibility of top event‘‘automatic gun cannot fire’’ based on data in Table 1 as the follows:
After the above calculation, it is shown that the failure possibil-ity of the top event ‘‘automatic gun cannot fire’’ is a fuzzy number(0.024742, 0.037396, 0.050861) and the reliability of the ‘‘auto-matic gun’’ is the fuzzy number. (0.949139, 0.962604, 0.975258).
4.4. Cheng et al.’s method
When system elementary event itself malfunction data isincomplete, Cheng et al. [24] proposed intuitionistic fuzzyfault-tree analysis procedure and method, integrating expert’sknowledge and experience in terms of providing the possibilityof failure of bottom events, and using triangle IFS to performcalculation, and then to determine the reliability interval forliquefied natural gas terminal emergency shutdown system.
Based on Cheng et al.’s method, calculations of the failurepossibility of top event ‘‘automatic gun cannot fire’’ based on dataof Table 1 are as follows:
According to Tω-based arithmetic operations on triangular IFSs(3)–(5), the failure range of top event ‘‘automatic gun cannot fire’’can be evaluated as:
According to Definition 3.1, calculating V (qT , qTj) as following.For obtaining the most critical bottom event of ‘‘automatic gun
cannot fire’’, we calculate the difference V (qT , qTj) (see Table 2).In conclusion, ‘‘Failure of magazine spring’’ (D) is the main reasonfor ‘‘automatic gun cannot fire’’ according to the ranking inTable 2. This is also the most significant factor of influence on gunfiring reliability. Therefore, at the managerial level, if we want toget higher reliability of gun firing, ‘‘Failure of magazine spring’’problem should take more concern. In other words, to improvereliability of gun firing, ‘‘Failure of magazine spring’’ problem ismore important than other bottom events.
5. Conclusion
In this paper, a new intuitionistic fuzzy fault-tree analysismodel based on weakest t-norm is proposed. This paper alsomodifies Tanaka et al.’s definition [4] on fault-tree analysisand integrates Tω-based arithmetic operations on triangular IFSsfor implementing fault-tree analysis on weapon system faultdiagnosis. The most critical system component (event ‘‘Failure ofmagazine spring’’) is the main reason for ‘‘automatic gun cannotfire’’. The proposed method is also compared with the existingtechniques of fault-tree methods. The summarized results of thesetechniques are listed in Table 3. Comparing the results of existingapproaches [2,4,24,25] and the proposed approach, it has beenshown that using the proposed approach, uncertainty about thereliability (length of reliability interval) decreases and the resultsare exact. While using the existing approaches the results areapproximate due to approximate product of triangular IFSs.
M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538 537
Table3
Compa
riso
nswith
othe
rfau
lt-tree
analysismetho
ds.
αFu
zzyfaulttree[4]
Crisp
possibility
[2]
Posb
ist[25
]Intuition
istic
fuzzyfault-tree
analysis[24]
Prop
osed
intuition
istic
fuzzyfault-tree
Leften
dpo
int
Middle
point
Righ
tend
point
a′a
bc
c′a′
ab
cc′
1.0
0.03
7396
0.03
7396
0.03
7396
0.03
7396
0.00
80.9
0.03
613
0.03
7396
0.03
8742
0.03
7396
0.00
80.8
0.03
4865
0.03
7396
0.04
0089
0.03
7396
0.00
80.7
0.03
360.03
7396
0.04
1435
0.03
7396
0.00
80.03
7396
0.03
7396
0.03
7396
0.03
7396
0.03
7396
0.03
7396
0.6
0.03
2334
0.03
7396
0.04
2782
0.03
7396
0.00
80.03
3901
0.03
7396
0.03
7396
0.03
7396
0.04
1086
0.03
6703
0.03
7396
0.03
7396
0.03
7396
0.03
8223
0.5
0.03
1069
0.03
7396
0.04
4128
0.03
7396
0.00
80.03
0407
0.03
5287
0.03
7396
0.03
964
0.04
4775
0.03
6009
0.03
6911
0.03
7396
0.03
7879
0.03
9051
0.4
0.02
9804
0.03
7396
0.04
5475
0.03
7396
0.00
80.02
6913
0.03
3178
0.03
7396
0.04
1884
0.04
8465
0.03
5316
0.03
6425
0.03
7396
0.03
8362
0.03
9878
0.3
0.02
8538
0.03
7396
0.04
6821
0.03
7396
0.00
80.02
3419
0.03
1069
0.03
7396
0.04
4128
0.05
2155
0.03
4623
0.03
594
0.03
7396
0.03
8845
0.04
0706
0.2
0.02
7273
0.03
7396
0.04
8168
0.03
7396
0.00
80.01
9925
0.02
896
0.03
7396
0.04
6372
0.05
5845
0.03
393
0.03
5455
0.03
7396
0.03
9329
0.04
1534
0.1
0.02
6008
0.03
7396
0.04
9514
0.03
7396
0.00
80.01
643
0.02
6851
0.03
7396
0.04
8617
0.05
9535
0.03
3237
0.03
497
0.03
7396
0.03
9812
0.04
2361
0.0
0.02
4742
0.03
7396
0.05
0861
0.03
7396
0.00
80.01
2936
0.02
4742
0.03
7396
0.05
0861
0.06
3225
0.03
2544
0.03
4485
0.03
7396
0.04
0295
0.04
3189
538 M. Kumar, S.P. Yadav / ISA Transactions 51 (2012) 531–538
Fig. 4. Membership function for top event.
References
[1] Ejlali A, Miremadi SG. FPGA-based Monte Carlo simulation for fault treeanalysis. Microelectronics Reliability 2004;44(6):1017–28.
[2] Kales P. Reliability: for technology, engineering, and management. Prentice-Hall; 1998.
[3] Zemva A, Zajc B. Test generation for technology-specific multi-faults based ondetectable perturbations. Microelectronics Reliability 2005;45:163–73.
[4] Tanaka H, Fan LT, Lai FS, Toguchi K. Fault-tree analysis by fuzzy probability.IEEE Transactions on Reliability 1983;32:150–63.
[5] Singer D. A fuzzy set approach to fault tree and reliability analysis. Fuzzy Setsand Systems 1990;34:145–55.
[6] Lee C, Lu TC, Lee NP, Chung UK. The study of strategy on new equipmentmaintenance. Fuzzy Sets and Mathematics 1999;13:37–44.
[7] Cai KY. System failure and fuzzy methodology: an introductory overview.Fuzzy Sets and Systems 1996;83:113–33.
[8] Tsujimura Y, Gen M. Fuzzy fault tree and its importance analysis. In:Proceeding of 16th ICC & IE. 1994. p. 301–4.
[9] Walley P. Statistical inferences based on second-order possibility distribution.International Journal of General Systems 1997;26:337–83.
[10] Suresh PV, Babar AK, Raj VV. Uncertainty in fault tree analysis: a fuzzyapproach. Fuzzy Sets and Systems 1996;83:135–41.
[11] Antonio CFG, Nelson FFE. Fuzzy FTA: a fuzzy fault tree system for uncertaintyanalysis. Annals of Nuclear Energy 1999;26:523–32.
[12] Atanassov KT. Intuitionistic fuzzy sets. Rep. no. 1697/84. Sofia (Bulgaria):Central Tech. Library, Bulgarian Academy Science; 1983.
[13] Biswas R. On fuzzy sets and intuitionistic fuzzy sets. Notes on IntuitionisticFuzzy Sets 1997;3:3–11.
[14] Bustince H, Burillo P. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets andSystems 1996;79:403–5.
[15] Atanassov K, Gargov G. Intuitionistic fuzzy logic. Comptes Rendus del’Académie Bulgare des Sciences 1990;43(3):9–12.
[16] Atanassov K, Georgeiv C. Intuitionistic fuzzy prolog. Fuzzy Sets and Systems1993;53:121–8.
[17] Li D-F. Multiattribute decision making models and methods using intuitionis-tic fuzzy sets. Journal of Computer and System Sciences 2005;70:73–85.
[18] Szmidt E, Kacprzyk J. Intuitionistic fuzzy sets in group decision making. Noteson Intuitionistic Fuzzy Sets 1996;2(1):11–4.
[19] De SK, Biswas R, Roy AR. An application of intuitionistic fuzzy sets in medicaldiagnosis. Fuzzy Sets and Systems 2001;117:209–13.
[20] Li DF, Cheng C. New similarity measures of intuitionistic fuzzy sets andapplication to pattern recognitions. Pattern Recognition Letters 2002;23:221–5.
[21] Chen SM. Analyzing fuzzy system reliability using vague set theory.International Journal of Applied Science and Engineering 2003;1(1):82–8.
[22] Shu MH, Cheng CH, Chang JR. Using intuitionistic fuzzy sets for fault-treeanalysis on printed circuit board assembly. Microelectronics Reliability 2006;46:2139–48.
[23] Chang JR, Chang KH, Liao SH, Cheng CH. The reliability of general vague faulttree analysis on weapon systems fault diagnosis. Soft Computing 2006;10:531–42.
[24] Cheng S-R, Lin B, Hsu B-M, Shu M-H. Fault-tree analysis for liquefied naturalgas terminal emergency shutdown system. Expert Systems with Applications2009;36:11918–24.
[25] Huang HZ, Tong X, Zuo MJ. Posbist fault tree analysis of coherent systems.Reliability Engineering and System Safety 2004;84:141–8.
[26] Zadeh LA. Fuzzy sets. Information and Control 1965;8:338–53.[27] Atanassov KT. Intuitionistic fuzzy sets. Rep. no. 1697/84. Sofia (Bulgaria):
Central Tech. Library, Bulgarian Academy Science; 1983.[28] AtanassovKT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems1986;20:87–96.[29] Chen SM. Measures of similarity between vague sets. Fuzzy Sets and Systems
1995;74:217–23.[30] Kumar A, Kumar S, Yadav SP. New approach for electric robot fuzzy
reliability analysis. International Journal of Performability Engineering 2007;3(2):257–66.
