ncetainties in Fault Tee nalysis Yue-Lung Cheng Depatment of Infomation Management Husan Chuang College 48 Husan-Chuang Rd. HsinChu Taiwan R.O.C bstact Fault tee analysis is one kind of the pobilistic safety analysis method. fte constucting a fault tee many basic events which can happen theoetically have neve occued so fa o have occued so infequently that thei easonle data ae not availle. Howeve the use of fuzzy pobility can descibe the failue pobility and its uncetainty of each basic event and then evaluate the pobility that the top event occus though cetain mathematical opeations. Howeve Guth [] has poposed using evidence theoy to pefom the fault tee analysis by the -valued logic. This pape shows that the lowe/uppe bound intevals obtained fom evidence theoy can be used to calculate the failue pobility inteval of the top event diectly i.e. without needing to tansfom into -valued foms. lthough some potion of the intevals may seen moe confident than othes diffeent kinds of membeship functions may be used to descibe subjective opinions while mathematical opeation can be pefomed to calculate the fault tee quantitative analysis. Key Wods: Fault tee analysis fuzzy numbe evidence theoy -valued logic membeship function. Intoduction Fault tee analysis was developed in 96 at ell Telephone Loatoies. Howeve cuent fault tee analysis still cannot be pefomed functionally without facing impecise failue input data and impope modeling poblems. Hence fuzzy sets which whee developed by L.. Zadeh thity yeas ago can help to ovecome this situation. Epets utilize fuzzy sets to subjectively descibe the uncetainties of each given event failue ate and then pefom mathematical opeation to evaluate system eliility. numbe of papes popose using fuzzy sets to descibe the impecision o vagueness of events in fault tee analysis [ 5 6 7 9 ]. Evidence theoy developed by Shafe [5] is a genealized ayesian statistics. Its main advantages ae: a the evidence theoy can obtain the lowe/uppe bound of each failue ate which can othewise be obtained by applying pobilistic techniques; and b diffeent opinions among epets can be pooled togethe to avoid bias coming fom some paticula epets. Guth [] fist poposed to utilize evidence theoy in fault tee analysis. He ewites the lowe/uppe bound of pobility as a -value logic and constucts ND/OR gates tuth tle to implement the fault tee quantitative evaluation. Ou contibutions in this pape ae: nlike in Guth's [] pape it is unnecessay to tansfom the intevals into -valued fom to calculate the failue ate of top event. Instead inteval calculations ae equivalent and much simple. the evidence theoy and fuzzy sets ae combined to evaluate fault tee quantitative analysis; and a fuzzy fault tee a evised multiplication fo the abitay shape membeship function is pesented.
. Fuzzy Fault Tee nalysis. Fault Tee nalysis fault tee is a gaphical epesentation of cetain elations which taces a system hazad backwads to seach fo all its possible causes. Such a system hazad is named as the top event of the fault tee. Taditionally quantitative analysis evaluates the pobility of the occuence of the top event in which case the pobility of each basic event is aleady known. Figue. is a fault tee eample which can also be descibed by the following elationship: T whee and Top Event 4 5 Figue. simple fault tee If i denotes the pobility of occuence of the event i the top event pobility would then be T { whee and I 4 } { 4 5 5 It is then easy to estimate the pobility of the top event if the pobility of each basic event is known as a cisp value. Howeve in most cases some basic events have neve o aely occued befoe poviding insufficient statistical estimation of pobilities. To ovecome this disadvantage fuzzy pobility was fist suggested by Tanaka [] to descibe the vague impecise phenomena fo the failue ates of the basic events.. Fuzzy Fault Tee nalysis } Consideing classical cisp sets fist Let E be a set and a subset of E then the elation between an element E and set is eithe o. Such a fact can be descibed o denoted by the following chaacteistic function. if µ 0 if whee µ is the chaacteistic function. Fo a fuzzy set sense the chaacteistic function may have diffeent degee of value say membeship between 0 and denote the belief that such as. Hence µ is often called the membeship function instead of chaacteistic function. If is a collection of objects denoted geneally by then a fuzzy set in is a set of odeed pais: { µ } whee µ is called the membeship function of in which maps to the membeship space [0]. Definition fuzzy subset is conve If μ min{ λ μ λ μ λ [ 0 } ] Definition fuzzy subset in R is nomal if and only if R µ ie. the highest value of µ is equal to one. In the fuzzy set theoy thee is a basic concept called etension pinciple which can be used to genealize cisp mathematical concepts to fuzzy sets. The etension pinciple is state as following: Definition Let be a catestin poduct of univeses L and L be fuzzy sets in L espectively. F is a mapping fom to a univese Y y L. Then the etension pinciple allows us to define a fuzzy set in Y by
{ y µ y y f L L } whee sup min{ µ L µ } L f y µ y if f y 0 0 othewise whee f is the invese of. Fuzzy obility The belief that an event is said to occu with pobility p can be descibed in tems of the following membeship function: if p µ p 0 othewise The belief that the event is said to occu with pobility p [ a b] can be descibed by the following ectangula membeship function: if [ a b] µ p 0 othewise The belief that the event occus with pobility p [ a b] in which some potion of [ a b] is moe accuate than othe potions can be descibed by a membeship function; the moe confident potion is given the value and othe potions ae given values between [ 0 ]. Epets can use diffeent kinds of membeship functions to subjectively addess the uncetainty of a failue pobility. Fuzzy pobility with tiangula [5] tapezoidal [] o bell-shaped [6] membeship function most often used fuzzy numbes and thei complements and appoimated multiplication both based on the etension pinciple. Fuzzy obility with abitay epesentation Within a fault tee not all fuzzy pobilities have to be epesented always by membeship functions of the same shape. Misa and Webe [8] povide a pocedue to implement the multiplication opeation fo fuzzy numbes of abitay shapes in the following steps: Step. Choose n sample points... n y y... y n with equal distance fo espectively such that n y yn ae lowe/uppe bounds of. Since C it is possible set z y z n n n fo lowe/uppe bounds of C and then divide [ z z n ] into n- subintevals of equal distance to obtain z z... zn. Kishna suggested that it may not be necessay to take n> unless thee ae many undulation in membeship function. n is usually an odd numbe Step. Choose a sample point such as of and egiste its membeship function value µ. Choose anothe sample point of the poduct set C fo eample a. point of can then be calculated by dividing a by i.e. a. Let this point be denoted by y. Then µ y can be easily obtained by fist locating two neaest sample points p p of and estimated µ p µ p µ y µ p y p p p by linea intepolation. Step. Once µ y is detemined fo such given sample points of the membeship function value µ a can be computed fom n C µ a µ i I µ a C i Step 4. Repeat step and step until all membeship value of sample points in C ae detemined. In this pape the multiplication in [8] was impoved by evising step slightly as follows : thee points ae chosen as the lowe/uppe bound points and nomal point and then the left pat and ight pat wee divided into subintevals of equal length espectively. In Kishna's method the nomal point may not be a sample. Hence it contadicts the idea that the fuzzy numbe must be nomal. The membeship functions ae easy to be used descibe the uncetainty of failue pobility fo a given basic event. Like the failue pobility is between 0.0 and 0. and is pehaps moe likely aound 0.07 see Figue.. Fuzzy membeship function gives moe pictuial imagine out the pobility it can be state the ange of the pobility distibution the moe confident potion the less confident y
potion. The pobility of the top event descibed by a fuzzy numbe instead of a cisp numbe can bing much moe infomation fo the manages to make decisions. elief function and plausibility function Let m be a given basic pobility assignment. function bel : θ [ 0] is called a belief function ove θ if and only if bel m. The plausibility function denoted as pls is a function pls : θ [ 0] defined by pls bel. The plausibility function pls may be epessed in tems of the basic pobility assignment m of bel as following : pls bel m m m θ I φ pdating pio mass Figue. Fuzzy obility. Evidence Theoy in Fault Tee nalysis The Dempste-Shafe theoy is based on the idea of placing a numbe between zeo and one to indicate the degee of evidence fo a poposition [0]. The theoy also includes easoning based on the ule of combination of degees of belief accoding to diffeent evidences. The addition aiom which states that fo any poposition in classical ayesian theoy does not necessaily coespond to the desciption of the eal wold because ignoance was not taken into account. Without enough evidence fo o against it is then appopiate to assume that the sum of both degees of belief ae not equal to one i.e. <. asic pobility numbes One of the basic concepts of the Dempste-Shafe theoy is that of a basic pobility assignment that is to assign a function m : θ [ 0] such that m0 0 m θ The numbe m is called a basic pobility numbe of. Condition states that no belief is committed to the empty set and states the total belief is equal to one. In the evidence theoy if additional infomation needs to be obtained these evidences may be pooled by Dempste's ule of combination Notation m pobility mass based on evidence m pobility mass based on evidence mc pobility mass fo the pooled evidence φ null set mφ mass assigned to φ Dempste's Rule computes mc as follows: 0 if C φ m m m m C I C othewise m m I φ -valued logic pplying the evidence theoy can obtain uppe and lowe bounds of a given component failue ate. In the fault tee analysis Guth [] suggested the -valued logic to manipulate the pobility inteval obtained fom evidence theoy. The belief that the event happens with failue pobility p [ bel pls ] means the event happens with pobility at least p bel at most p pls. Thee is an uncetainty ange with pobility pls bel. Hence Tue FalseF
nknown F ae defined as follows. m T bel ; m F pls and m F pls bel also m T m F m F Notation { a a a} Numbes denoting m ove the possibilities { T F F} whee a a a { b b } Numbes denoting m ove the possibilities { T F F} wheeb b b. Tle. Tuth tle fo the OR gate OR Tb Fb F b Ta Ta b Ta b Ta b Fa Ta b Fa b F a b F a Ta b F a b F a b Tle. Tuth tle fo the ND gate ND Tb Fb F b Ta Ta b Fa b Fa b Fa Fa b Fa b F a b F a Fa b F a b F a b Fom Tle. -valued logic OR of and is assigned the mass: m { m T m F m F} a b a a b b a a a b a b a b a b a b a Similaly fom Tle. -valued logic ND of and is assigned the mass: m I { m T m F m F} a b a a b a b b ab a b a b a a b a b a b a Ou impovement based on inteval calculations In the pevious section Guth's method tansfomed [bel pls] into a -valued patten. Now anothe method is used to teat the belief and plausible [4]. Definition Let I { α β α β [0]}. Two opeatos call inteval addition and inteval multiplication espectively ae defined as follows: Fo I α β I α β I I I α β α α α β I I α β α αα ββ and its complement is defined as: I α β α Theoem The calculation of intevals and -valued logic ae equivalent. oof: Suppose m a a and a m b b fo the event espectively. Now we ewite events in the fom of inteval. : a a a α : b b b α and so p p β β β β α α α α α α α α β β β a b a b a a b a a b a ab ab a a a b b α since sing -valued logic to intepet the esult ove: m T α a a b a b β α ab β a a m F a b m F b
and fo I p p α β α α β α β a b a a b a b ab a α I I sing -valued logic to intepet the esult ove: m T α a b I α I β I ab a a b a m F a b m F a b The final esult coesponds to that of Guth. Q.E.D. E TO EVENT E E Figue.. simple fault tee Eample s the fault tee in Figue. pobility inteval fo event E E E ae 0.0. 0.0. 0.0.4 then what is the pobility inteval of the top event? Guth's Method: t fist we tansfom the intevals into -valued foms then we have {0.0.80.} {0.0.70.} {0.0.60.} fo E E E espectively. Since TO { E E I E apply equation & we get that mto {0.0840.840.09} Ou Method: We use the lowe/uppe bounds diectly and manipulate the inteval calculation and get m TO 0.0840.76. We get the same esult as Guth's method. 4. Conclusion In this pape inteval calculations wee used to manipulate the lowe/uppe intevals [bel pls] instead of tansfoming them into -valued fom. y a simple compaison it is easy to see the inteval calculation is moe concise and efficient in opeation. The inteval may also be viewed as ectangula shape fuzzy numbe which means that thee is no weight upon the inteval. Once the analyst has his o he subject confidence upon the inteval diffeent kinds of membeship functions can be constucted within the inteval. Fuzzy sets and the evidence theoy ae two main methods to descibe uncetainty both having thei own advantage in manipulating uncetainty. ppopiate combination of the two methods is the main idea in this pape. Refeence [] Cheng C. H. and Mon D. L. Fuzzy system eliility analysis by inteval of confidence Fuzzy Sets and Systems Vol.56 pp. 9-5 99 [] Chen L. C. and Yuan J. Studies on combining fuzzy sets and evidence theoy in fault tee analysis Depatment of Industial Engineeing National Tsing Hua nivesity June 994 [] Guth M.. pobility foundation fo vagueness and impecision in fault tee analysis IEEE Tans. Reliility Vol.40 No.5 pp. 56-570 99 [4] Kaufman. and Gupta M. M. Intoduction to Fuzzy ithmetic Theoy and pplications Van Nostand Reinhold 99 [5] Liang G. S. and Wang M. J. Fuzzy fault tee analysis using failue possibility Micoelecton Reliility Vol. Vol.4 pp. 58-597 99 [6] Liao W. H. and Yuan J. epet system fo fuzzy fault tee application and fault tee Diagnosis Depatment of Industial Engineeing National Tsing Hua nivesity June 99 [7] Misa K.. and Wede G. G. new method fo fuzzy fault tee analysis Micoelecton Reliility Vol.9 No. pp. 95-6 989 [8] Misa K.. and Webe G. G. se of fuzzy set theoy fo level I studies in
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