causes of uncertainty in fta method and use of...

American Journal of Oil and Chemical Technologies; ISSN (online): 2326-6589; ISSN (print): 2326-6570

Volume 2, Issue 5, May 2014

150

Causes of Uncertainty in FTA Method and Use of Fuzzy Logic to Solve this Problem

Kh . Seiah Mansour1, N. Nabhani 1 , B. Anvaripour1 , F. Jaderi2

1. Abadan Institute of Technology, Petroleum University of Technology, Abadan, Iran 2. Faculty of Environmental Studies, University Putra Malaysia, Malaysia

Corresponding Author Email: [email protected]

Abstract:

Nowadays increasing the number of chemical industries leads to more industrial accidents. Due to consequences of accidents imposed high costs to industry, society and environment. This makes safety analysis more important. Therefore qualitative and quantitative hazard analyses are essential for identification and quantification of hazards. Fault tree analysis (FTA) is an established technique in hazard identification. People when using FTA often suffers from a lack of detailed data on failure rates, uncertainties in available data, imprecision and vagueness. This may lead to uncertainty in results and process risk level. Fuzzy logic deals with uncertainty and imprecision, and is an efficient tool for solving problems where knowledge of uncertainty may occur. In this paper traditional FTA was combined with fuzzy set theory. In fuzzy method, all variables are replaced by fuzzy numbers in the process of fuzzification and subsequently fuzzy probability of the top event is used for fault tree.

Keyword: Fault Tree Analysis, Uncertainty, Fuzzy Logic, Fuzzy Operator

1. Introduction

Fault Tree Analysis (FTA) is one of the most important logic and probabilistic techniques used in Probabilistic Risk Assessment (PRA) and system reliability assessment today. FTA is a deductive process that can be simply described as an analytical technique, whereby an undesired state of the system is specified (usually a state that is critical from a safety or reliability standpoint), and the system is then analyzed to find all realistic ways in which the undesired event (top event) can occur. Indeed FTA splits up a system failure into more detailed events, such as subsystem failures, that might be responsible for this occurrence. This process is repeated for each new event found and proceeds until only basic events lacking more detailed descriptions are left. The fault tree itself is a graphic model of the various parallel and sequential combinations of faults that will result in the occurrence of the predefined undesired event. The faults can be events that are associated with component hardware failures, human errors,



151

software errors, or any other related events which can lead to the undesired event. A fault tree thus describes the logical interrelationships of basic events that lead to the undesired event [1].

The fault tree analysis technique was first developed in 1961-1962 by H.A. Watson of Bell Telephone Laboratories in connection with an U.S. Air force to facilitate analysis of the launch control system of the intercontinental Minuteman missile. Dave Haasl, then at the Boeing Company, recognized the value of this tool and led a team that applied FTA to the entire minuteman missile system. Other divisions within Boeing saw the results from the Minuteman program and began using FTA during the design of commercial aircraft. In 1965 Boeing and University of Washington sponsored the first system safety conference. At this conference, several papers were presented on FTA, marking the beginning of worldwide interest in FTA.

Table1. features of fault tree analysis

Major applications of FTA Major industries and technologies utilizing FTA

Benefits of Constructing a Fault Tree

1 Numerical requirement verification

1 Aircraft-commercials, fighters, bombers, tankers, helicopters

1

The fault tree explicitly shows all the different relationships that are necessary to result in the top event.

2

Identification of safety critical components

2 Power systems-nuclear, solar, electric

3 Product certification 3 Transit systems- trains, BART

2

In constructing the fault tree, a thorough understanding is obtained of the logic and basic causes leading to the top event.

4 Product risk assessment 4 Space-Apollo, space shuttle, satellites, launch vehicles, space station

5 Accident/incident analysis

5 Robotic systems 3

The fault tree is a tangible record of the systematic analysis of the logic and basic causes leading to the top event.

6 Design change evaluation

6 Auto systems

7 Visual diagrams of cause-consequence events

7 Oil platforms 4

The fault tree provides a framework for thorough qualitative and quantitative evaluation of the top event

8 Common cause analysis 8 Torpedoes 9 Hydrofoil



152

The traditional FTA based on probabilistic approach is widely use in the past. Usually it is very difficult to estimate precise failure probabilities and failure rates of each components or failure events. Especially this occurs in systems such as nuclear power plants where available data are not enough for statistical inferences or the data displays a wide variation. Hence, in the absence of accurate data, it might be necessary to use rough estimates of probabilities. The failure rates or failure probabilities are treated as random variables with known probability distributions to integrate the variation in the estimated values. This requires that data be available from which these probability distributions can be reasonably inferred. In the traditional uncertainty analysis, the point estimates of the primary events are replaced by probability distributions. Therefore a probability distribution for the probability of occurrence of the top event in the fault tree can be derived. But an analytical method may be difficult and a simulation may need very much computer time. Fuzzy methods may be the only resort when little quantitative information is available regarding fluctuations in the parameters. In fuzzy approach the algebraic operations are easy [2].

First research on the Fuzzy Fault Tree Analysis (FFTA) was done by Tanaka et al. in 1983. They treated probabilities of basic events (BEs) as trapezoidal fuzzy numbers, and applied the fuzzy extension principle to determine the probability of Top Event. They defined an index function analogous to importance measures for evaluating to what extent a basic event contributes to the top event [3]. In 1984, Furuta and Shiraishi proposed the concept of fuzzy importance as a correspondence to probability-based importance. They used representative values of fuzzy membership functions to calculate its importance [4]. In 1986, linguistic variables are introduced by Waldemar Karwowski and Anil Mital to analyze potentially hazardous situations using approximate reasoning methods [5]. In 1990, D. Singer used fuzzy set theory to replace crisp numbers by fuzzy numbers for better estimation of possibility of top event in FTA [6, 7]. Cheng and Mon (1993) in order to simplify the calculation of Singer's method suggested revised methods to analyses fault trees by specifically considering the failure probability of basic events as triangular fuzzy numbers [8]. Lin and Wang (1997) developed a hybrid method which can simultaneously deal with probability and possibility measures in a FTA. In this approach, the hardware failure rates are still based on probabilistic data. In addition, they use fuzzy possibility score to represent the failure occurrence of the fuzzy events such as human errors. The evaluation data were expressed in linguistic terms, e.g. fairly low, low, and very low, etc. The triangular and trapezoidal fuzzy numbers were used to denote failure possibilities. In order to integrate the hardware failure rate and the fuzzy possibility score of human error, a transformation function is used to convert fuzzy possibility score into fuzzy failure rate [9]. Onisawa (1988) proposed a method of using the concept error possibility instead of the error rate in traditional reliability analysis and presented a method of fault tree analysis of human errors based on this concept [10]. Huang et al. (2004) adopted possibility theory to analysis fuzzy fault trees [11]. Dong and Yu (2005) applied the hybrid method to analyses failure probability of oil



153

and gas transmission pipeline. They were treated the probabilities of basic events of oil and gas transmission pipeline as fuzzy number, which could be obtained by expert elicitation and theory of fuzzy set and failure probabilities of the top event and fuzzy failure probabilities of the basic events is used to evaluate important analysis of basic events [12]. Ping et al. (2007) presented a method that overcomes the weaknesses of traditional FTA by using possibilistic measures and fuzzy logic. They created the fuzzy logic-based possibilistic FTA model with four steps. Possibility of failure, analogous to probability of failure in probabilistic FTA, is first defined, and then fuzzy variables are showed describing sub-events in a natural way. After introducing an importance measure, fuzzy rules are generated from linguistic quantification and meaning inference in fuzzy logic [13]. Adam S. Markowski et al. in 2009 presented the application of fuzzy sets theory as a basic tools used in process safety analysis such as fault and event tree methods and they showed that it can be used in the ‘‘bow-tie’’ approach for accident [14]. Huang et al. in 2012 used fuzzy fault tree analysis to uncontained events of an Aero-Engine rotor [15].

2. Basics of Fuzzy Set Theory

Fuzzy set theory is a generalization of ordinary or classical set theory; it consists of mathematical tools developed to model and process incomplete and/or gradual information, ranging from interval-valued numerical data to symbolic and linguistic expressions. The concepts and definitions of the fuzzy set theory are briefly summarized below.

2.1 Fuzzy Set and its Membership Function

A fuzzy set is a class of objects with a continuum of grades of membership. A fuzzy set is characterized by a membership function which assigns to each object a grade of membership ranging between zero and one. Unlike crisp (or ordinary) sets, fuzzy sets have no sharp or precise boundaries.

Let X be a space of points (objects), with a generic element of X denoted by x. Thus, X={x}. A fuzzy set Ã in X is characterized by a membership function µA(x) which associates with each point in X a real number in the interval [0, 1], with the value of µA(x) at x representing the "grade of membership" of x in Ã [16].

The membership function µA(x)= 1 represents elements that are completely in Ã, 0 represents elements that are completely not in Ã, and values between 0 and 1 represent partial inclusion in A. Formally, Ã is represented as the ordered pair (x, µA(x) ). The degree of membership µA(x) of elements x to a fuzzy set Ã depends on their positions or compatibility with respect to the Centre (or the central concept) [18]. The definition of a fuzzy set depends on the situation (or context) that modifies our perception or expectation of the meaning of the linguistic variable.



154

2.2 Extension Principle

The extension principle is used to extend crisp mathematical concepts to fuzzy sets. Following Lotfi Zadeh it is defined by Zimmermann as follows:

Let X be a Cartesian product of universes X = XI x X2 x ... x Xr and let Ã1, Ã2, …, Ãr be r fuzzy sets in X1 , . . . , Xr respectively. If f is a mapping from X to a universe Y, y=f(x1,..., xr) then the fuzzy set B in Y is defined by:

B = {(y,µB(y)) | y = f(x1,...,xr), (x1,...,xr) ϵ X} ( 1)

In the simplest case when f maps a single fuzzy set A in a universe X into a universe Y(r = l), (1) becomes

Max x= f-1(y) µ(x) , if f-¹(y) ≠Ø

µB(y) = (2)

0 otherwise [18]

2.3 Analytical Representation of Membership Functions:

Membership functions can be represented by different methods. One of these methods, analytical representation is used when a fuzzy set is defined over a continuous domain, such as real numbers or gradational classes of objects, which makes it impossible to represent all the elements x with their membership functions µA(x) in any other form.

The membership function, µA(x) is defined by the typical convex functions of triangular, trapezoidal and Gaussian type. The selection of a membership function shape depends on the characteristics of variables. In majority cases the shape of the membership function does not affect essentially the final result [17, 18].

Fuzzy numbers:

A fuzzy set Ã defined on the universal set of real numbers R is said to be a fuzzy number if its membership function has the following characteristics:

a. µA (x): R → [0; 1] is continuous. b. µA (x) = 0 for all x∈ (-∞, c] ∪ [d, ∞), where c<d. c. µA(x) is strictly increasing on [c, a] and strictly decreasing on [b, d] for c ≤ a ≤ b ≤ d. d. µA (x) = 1 for all x∈ [a, b] provided a ≤ b. [20]



155

Triangular fuzzy numbers:

A fuzzy number A is termed as triangular fuzzy number (Figure. 1) if the membership function of fuzzy number A is defined by the following expression:

0 if x ≤ a

µA(x) = !!!!!!

if a ≤ x ≤m (3)

!!!!!!

if m ≤ x ≤b , x, a ,m, bϵ R [20]

A triplet (a, m, b) may be used to denote each triangular fuzzy number defined above.

Trapezoidal fuzzy numbers:

A trapezoidal fuzzy number (Figure. 2) A denoted by a quadruple (a1, a2, a3, a4) can be defined as follows:

0 if x ≤ a1 or x≥a3

µA(x) = !!!!!!!!!

if a1 ≤ x ≤a2

1 if a2 ≤ x ≤a3 (4)

𝒂𝟒!𝒙𝒂𝟒!𝒂𝟑

if a3 ≤ x ≤a4 x, a1 ,a2, a3,a4 ϵ R [20]

Figure1. Triangular fuzzy number [20] Figure 2. Trapezoidal fuzzy number [20]



156

The union (AUB) and intersection (A∩B) of two fuzzy sets A and B are given, respectively by:

µAUB(x) = max (µA(x), µB(x)) (5)

µA∩B(x) = min (µA(x), µB(x)) (6)

The addition of triangular fuzzy number A = (a1, a2, a3) and B = (b1, b2, b3) is defined as:

A+ B = (a1 + b1, a2 + b2, a3 + b3) (7)

Thus the addition of two triangular fuzzy numbers is again a triangular fuzzy number. Similarly subtraction of two triangular fuzzy numbers is also a triangular fuzzy number and it can be given by the following expressions:

A− B = (a1 –b1, a2 – b2, a3 – b3) (8)

The multiplication of two fuzzy numbers A= (a1, a2, a3) and B = (b1, b2, b3) denoted as A*B can be defined as:

-D1+ [D12+(x-P)/T1]½ P≤ x ≤Q

µA*B(x) = -D1- [D22+ (x-R)/ U1]½ Q≤ x ≤R (9)

0 otherwise

where T1=(a2-a1)(b2-b1), T2=a1(a2-a1)+b2(b2-b1), U1=(a2-a1)(b2-b1), U2=b3(a2-a1)+a3(b2-b1), D1=T2⁄(2T1) , D2=-U2/(2U1) , P=a1b1 , Q=a2b2, R= a3b3 .

The fuzzy number A*B is not a triangular fuzzy number. But in most of the cases, computation with resulting fuzzy numbers becomes very tedious. Thus it is necessary to avoid the second and higher degree terms to make them computationally easy and therefore the product of two fuzzy numbers is reduced to a triangular fuzzy number (P, Q, R) or (a1b1, a2b2, a3b3). With a similar approach one can define the algebraic operations on trapezoidal fuzzy numbers. Thus the addition, subtraction and multiplication of two trapezoidal fuzzy numbers A= (a1, a2, a3, a4) and B= (b1, b2, b3, b4) are again represented by the trapezoidal fuzzy numbers (a1+b1, a2+b2, a3+b3, a4+b4), (a1-b1, a2-b2, a3-b3, a4-b4) and (a1b1, a2b2, a3b3, a4b4) respectively.

For α ϵ [0, 1], the α-level cut of Ã (simply α-cut) is defined by: [Ã]α={x | x ϵ R, µA(x) ≥α}.



157

2.4 Defuzzification Methods

Defuzzification is a process of reducing an aggregated (or clipped) fuzzy set into a crisp number, presumably the most representative value of that fuzzy set interval. Defuzzification is the process of producing a quantifiable result in fuzzy logic.

Defuzzification of triangular fuzzy number Ã = (a1, a2, a3) by Centre of area method is:

X*=!!!!!!!!

!!!! !"!! !!!!

!!!!!!!!! !"!

!!!!!!!!

!!!! !"! !!!!

!!!!!!!!! !"

= (!!)(𝑎1+ 𝑎2+ 𝑎3) (10)

Defuzzification of trapezoidal fuzzy number Ã = (a1, a2, a3, a4) can be obtained by Equation below.

X*=!!!!!!!!

!!!! !"!! !"!!!

!! ! !!!!!!!!!

!!!! !"!

!!!!!!!!

!!!! !"! !"!!

!! ! !!!!!!!!!

!!!! !"

= (!!)( !!!!!

!!!!!!! !!!!! !!!!!!(!!!!!!!!!!!)

) (11)

3. The Procedure to Fuzzy Fault Tree Analysis (FFTA)

FTA is a deductive technique where start with a failure scenario being considered, and decompose the failure reasons into its possible causes. Each possible cause is then investigated and further refined until the basic causes of the failure are understood. The failure scenario to be analyzed is normally called the TOP event of the fault tree. The basic causes are the basic events of the fault tree. The fault tree should be completed in levels, and they should be built from top to bottom.

3.1 Construct the FT

In FTA first the boundary of the system under study is defined and then a particular system failure for more analysis is selected. After that the top event must be defined. The analyst next determines the immediate, necessary, and sufficient causes for the occurrence of this top event. It should be noted that these are not the basic causes of the event but the immediate causes. This is an extremely important point. Draw the appropriate gate to describe the logic for these events resulting in the top event. The immediate, necessary, and sufficient causes of the top event are now treated as sub-top events and the analyst then causes. In so doing, the analyst is placed in the



158

position of the subsystems analyst for whom the failure mechanisms are the failure modes; that is, the sub-top events correspond to the top events in the subsystem fault tree [1]. Three types of symbols can be distinguished: event symbols, gate symbols and transfer symbols.

Event symbols:

The following event symbols are commonly used:

• A circle represents a basic event such as the failure of an elementary component for which sufficient probability information is available and further development is not required.

• A diamond expresses an undeveloped or incomplete event which is not developed further due to lack of time or interest, or because more detailed information is not available.

• A rectangle signifies a gate output event or resulting event which results from the combination of input events through the corresponding gate. The top event itself is a resulting event as well.

• An oval symbol denotes a conditional event. It indicates any condition or restriction that applies to a logic gate.

• The house-shaped symbol indicates a trigger event, i.e. an event which is expected (or simulated to occur.

Gate symbols:

Gate symbols connect events according to their causal relations. There are two basic types of fault tree gates, the OR-gate and the AND-gate. All other gates are special cases of these two basic types. With one exception, gates are symbolized by a shield with a flat or curved base. The most important gates are:

• The AND-gate which indicates that an output event occurs if and only if all the input events occur simultaneously (Figure. 3).

• The OR-gate is used to show that the output event occurs only if one or more of the input events occur. There may be any number of input events to an OR-gate and is shown by Figure 4.

Figure3. AND-gate [1] Figure 4. OR-gate [1]

Transfer symbols:



159

The triangles are introduced as transfer symbols and are used as a matter of convenience to avoid extensive duplication in a fault tree or to allow a large tree to be represented on a number or smaller trees for clarity. A “transfer in” gate (Fig. 5) will link to its corresponding “transfer out”. This “transfer out” gate (Figure. 6) perhaps on another sheet of paper, will contain a further portion of the tree describing input to the gate [1].

Figure5. Transfer IN [1] Figure 6. Transfer OUT [1]

3.2 Evaluation the FT

Depending on the objectives of the analysis, FTA can be qualitative or quantitative. In the following possible results and analysis methods for qualitative and quantitative FTA will be discussed in detail.

3.2.1 Qualitative Analysis

The qualitative evaluation usually provides information on the minimal cut sets for the top event. A cut set in a fault tree is a set of basic events whose occurrence leads to the occurrence of the top event. The nature of the basic events and the number of basic events in the combined sets give important information about the top event occurrence.A minimal cut set is a smallest combination of component failures which, if they all occur, will cause the top event to occur. By the definition, a minimal cut set is a combination (intersection) of primary (basic) events sufficient for the top event.

3.2.2 Quantitative Analysis

Quantitative analysis is used to evaluation of failure probability of the top event and important analysis of the basic events. If all the MCSs and probabilities of the basic events were obtained, the failure probability of the top event would be achieved. Important analysis of the basic events is an important part of the quantitative analysis. It indicates contribution of the basic events to the failure of top event [12].

Final calculation of fuzzy probability of top event (FPTE) follows an appropriate equation for minimal cut set (MCS) for particular structure of the fault tree.



160

FPTE = (FPBE1 •…• FPBEi )1+ … + (FPBE1•…• FPBE k) n (12)

FPBEi :fuzzy probability of basic event "i''.

Using algebraic operations on fuzzy numbers (triangular or trapezoidal) we can obtain fuzzy operators FNOT, ANDF and ORF corresponding to Boolean operators "NOT", "AND" and "OR" respectively as follows.

(i) If a fuzzy event ‘i’ is represented by a possibility function Pi the generalized Boolean operator NOT to be denoted by FNOT and defined as:

FNOT pi =1− pi =1− (ai1, ai2, ai3) = (1− ai3, 1− ai2, 1− ai1) (for triangular fuzzy numbers) (13)

FNOT pi =1− pi =1− (ai1, ai2, ai3, ai4) = (1− ai4, 1− ai3, 1− ai2, 1− ai1) (for trapezoidal fuzzy numbers) (14)

(ii) If P1, P2 …, Pn are the possibility functions of n basic events and Py be the same for resulting event. Then the fuzzy operators ANDF and ORF are defined in the following manner:

Py = ANDF (P1, P2…..Pn) = p!!!!! , (15)

Where Π denotes the fuzzy multiplication and Py be the possibility of resulting event.

Let Pi’s are represented by triangular fuzzy numbers (ai1, ai2, ai3), i=1, 2…n, then

P y = ANDF( P1 , P2 ,… ,Pn ) = (𝑎𝑖1 , 𝑎𝑖2 , 𝑎𝑖3!!!!! ) = ( a!"!

!!! , a!"!!!! , a!"!

!!! ) (16)

Similarly, if Pi’s are assigned trapezoidal fuzzy numbers (ai1, ai2, ai3, ai4), i=1, 2…n, then

P y = ANDF( P1 , P2 ,…,Pn ) = (𝑎𝑖1 , 𝑎𝑖2 , 𝑎𝑖3, 𝑎𝑖4)!!!! = ( a!"!

!!! , a!"!!!! , a!"!

!!! , a!"!!!! ) (17)

Let Pi’s are triangular fuzzy numbers (ai1, ai2, ai3), i=1, 2…n, then

P y = ORF (P1, P2… Pn) = 1- (1 − ai1, ai2, ai3 )!!!! = 1- ( (1 − a!" , 1 −!

!!! a!", 1 − a!")) = 1- ( (1 −!!!!

a!"), (1 − a!")!!!! , (1 − a!")!

!!! ) = (1- (1 − a!")!!!! , 1- (1 − a!")!

!!! , 1- (1 − a!")!!!! ) (18)

Similarly, if Pi’s are trapezoidal fuzzy numbers (ai1, ai2, ai3, ai4), i=1, 2…n, then

P y = ORF (P1, P2… Pn) = 1- (1 − ai1, ai2, ai3 , ai4 )!!!! = (1- (1 − a!")!

!!! , 1- (1 − a!")!!!! , 1- (1 −!

!!!

a!"), 1 - (1 − a!")!!!! ) (19)

When minimal cut sets were determined by using fuzzy operators that defined above and fuzzy probability of basic events probability of top event can be calculated.



161

4. Conclusion

The major purpose of this paper is to present a fuzzy method for fault tree analysis. A fuzzy methodology for fault tree evaluation seems to be a viable alternative solution to overcome the weak points of the conventional approach: insufficient information concerning the occurrences frequencies of hazardous events. This approach is more straight-forward compared to another methods.

References

[1] M. Stamatelatos, NASA HQ, OSMA, Fault Tree Handbook with Aerospace Applications, NASA Office of Safety and Mission Assurance NASA Headquarters, Washington, DC, 2002.

[2] D. Dubois, Francesc Esteva, Llu´ıs Godo, Henri Prade, Fuzzy-Set based logics - a history-oriented presentation of their main developments, Handbook of the History of Logic, Volume 8, Elsevier, 2007.

[3] H. Tanaka, L.T. Fan, F.S. Lai and K.Toguchi, Fault-tree analysis by fuzzy probability. IEEE Trans. Reliability, Vol. 32, pp. 453-457, 1983.

[4] H. Furuta and N.Shiraishi. Fuzzy importance in fault tree analysis. Fuzzy Sets and Systems, Vol. 12, pp. 205-213, 1984.

[5] W. Karwowski and A. Mital, Potential application of fuzzy sets in industrial safety engineering, Fuzzy Sets and Systems, 19, 105–120, 1986.

[6] D. Singer, A fuzzy set approach to fault tree analysis, Fuzzy Sets and Systems 34, pp.145-155, 1990.

[7] D. Singer, Fault tree analysis based on fuzzy logic, Computers & Chemical Engineering, Vol. 14, No. 3, PP. 259-264, 1990.

[8] C. Ching-Hsue and M. Don-Lin, Fuzzy system reliability analysis of confidence, Fuzzy Sets and Systems, 56, pp. 29-35, 1993.

[9] L. Ching-Torng and J. W. Mao-Jiun, Hybrid fault tree analysis using fuzzy sets, Reliability Engineering and System Safety, 58, pp.205-213, 1997.

[10] O. Takehisa, An approach to human reliability in man-machine systems using error possibility, Fuzzy sets and systems, 27, pp.87-103, 1988.

[11] H. Hong-Zhong, T. Xin , Ming J. Zuo, Posbist fault tree analysis of coherent systems, Reliability Engineering and System Safety, 84, pp. 141–148, 2004.

[12] Y. Dong, D. Yu, Estimation of failure probability of oil and gas transmission pipelines by fuzzy fault tree analysis, Journal of Loss Prevention in the Process Industries, 18, pp. 83–88, 2005.

[13] H. Li-Ping He, H. Hong-Zhong, Ming J. Zuo, Fault Tree Analysis Based on Fuzzy Logic, IEEE Trans, Reliability, pp. 453-457, 2007.

[14] S. Adam, a. Markowski, M. Sam Mannan b, B. Agata, Fuzzy logic for process safety analysis, Journal of Loss Prevention in the Process Industries, 22, pp.695–702, 2009.



162

[15] L. Yanfeng, H. Hong-Zhong, Z. Shun-peng, L. Yu, An application of fuzzy fault tree analysis to un contained events of an Areo- engine Rotor, Int.J. Turbo Jet-Engines, 29, pp. 309-315, 2012.

[16] L.A. Zadeh, Fuzzy sets: Information and control, Vol. 8, pp. 338-353, 1965. [17] Johen B. Bowles, Colon E. Pelaez, Application of Fuzzy Logic to Reliability

Engineering, Proceeding of the IEEE, VOL. 83, and NO. 3, pp.435-449, 1994. [18] A. Aydin, Fuzzy set approaches to classification of rock masses, Engineering

Geology, 74, Issues 3-4, pp. 227-245, 2004. [19] H. Nasseri, Fuzzy Numbers: Positive and Nonnegative, International

Mathematical Forum, 3, no. 36, pp.1777 – 1780, 2008.

causes of uncertainty in fta method and use of...

Documents