Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) Aalysis of Fuzzy Fault Tree usig Ituitiostic Fuzzy Numbers Neeraj Lata Departmet of applied Scieces,TeerthakerMahaveer Uiversity, Moradabad(U) mail id: sirohiphysics@rediffmail.com Abstract I geerally fuzzy seats are used to aalyses the system reliability. I preset paper we have preseted a ew approach to evaluate the reliability of fuzzy fault tree usig Ituitiostic Fuzzy Numbers. I applicatio of Ituitiostic Fuzzy fault tree aalyses, we have itroduced a ew distace methods betwee two iterwal estimates of reliability with truth values. Usig this methods, the importace idex is calculated ad compared with the weighted idex. KEYWDS: Ituitiostic Fuzzy Numbers, Fault Tree Model.. Itroductio The cocept of fault tree aalysis (FTA) was developed i 96 at Bell telephoe laboratories. FTA is ow widely used i may fields, such as i uclear reactor, chemical ad aviatio idustries. Fault tree aalysis (FTA) is a logical ad diagrammatic method for evaluatig system reliability. It is logical approach for systematically quatifyig the possibility of abormal system evet. Startig from the top evet the fault tree method employs Boolea algebra ad logical modelig to represet the relatios amog various failure evets at differet levels of system decompositio. FTA ca be a qualitative evaluatio or quatitative aalysis. However, curret fault tree aalysis still caot be performed fuctioally without facig imprecise failure ad improper modelig problems. FTA is ow widely used i may fields such as i the uclear reactor ad chemical idustries. The reliability of a system is the probability that the system will perform a specified fuctio satisfactorily durig some iterval of time uder specified operatig coditios. Traditioally, the reliability of a system behaviour is fully characterized i the cotext of probability measures, ad the outcome of the top evet is certai ad precise as log as the assigmet of basic evets are descet from reliable iformatio. However i real life systems, the iformatio may be iaccurate or might have liguistic represetatio. I such cases the estimatio of precise values of probability becomes very difficult. I order to hadle this situatio, fuzzy approach is used to evaluate the failure rate status. Fuzzy fault tree aalysis has bee used by several researchers[5,6,7,,] ad Siger[] proposed a method usig fuzzy umbers to represet the relative frequecies of the basic evets. He used possibilistic AND, ad NEG operators to costruct possible fault tree.. Basic Notios ad Defiitios of Ituitioistic Fuzzy Sets (IFS s ):- Fuzzy set theory was first itroduced by Zadeh i 965[3]. Let X be uiverse of discourse defied by X = {x, x,...,x }. The grade of membership of a elemet x i X i a fuzzy set is represeted by real value betwee 0 ad. It does idicate the evidece for x i X, but does ot idicate the evidece agaist x i X. Ataassov i 984[,] preseted the cocept of IFS, ad poited out that this sigle value combies the evidece for x i X ad the evidece agaist x i X. A IFS A i X is characterized by a membership fuctio μ A (x) ad a o membership fuctio ν A (x). Here μ A (x) ad ν A (x) are associated with each poit i X, a real umber [0 ] with the value of μ A (x) ad ν A (x) at x represetig the grade of membership ad o membership of x ia. Thus closure the value of μ A (x) to uity ad the value of ν A (x) to zero; higher the grade of membership ad lower the grade of o-membership of X. Whe A is a crisp set, its membership fuctio (omembership) ca take oly two values zero ad. If μ A (x) = ad ν A (x) = 0, the elemet x belogs to A. Similarly if μ A (x) =0 ad ν A (x) =, the elemet does ot belogs to A. A IFS becomes a fuzzy set A whe ν A (x) = 0 but μ A (x) [0 ], x A.. Defiitio of Ituitioistic Fuzzy Set: - Let E be a fixed set. A Ituitioistic fuzzy set A of E is a object havig the form A = {< x, μ A (x), ν A (x) >: x E} Where the fuctios ISSN : 9-3345 Vol. 4 No. 07 Jul 03 98
Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) μ A : E [0 ] ad ν A : E [0 ] defie respectively, the degree of membership ad the degree of o-membership of the elemet x Eto the set A, which is a subset of E ad for every x E, 0 μ A (x) + ν A (x). Whe the uiverse of discourse E is discrete, a IFS A ca be writte as A = [μ A (x), ν A (x)]/x, x i E A IFS A with cotiuous uiverse of discourse E ca be writte as A = [μ A (x), ν A (x)]/x, x i E E Fig. Membership ad o-membership fuctios of A. Triagular Ituitioistic Fuzzy Numbers (TIFN):- The TIFN A is a Ituitioistic Fuzzy umber (A ) is a Ituitioistic Fuzzy set i R with five real umbers (a, a, a 3, a, a ) with (a a a a 3 a ) ad two triagular fuctios x a, for a a a x a μ A (x) = a 3 x, for a a 3 a x a 3 0, otherwise a x, for a x a a a ν A (x) = x a, for a a a x a, otherwise Fig. Membership ad o-membership fuctios of TIFN.3 Existig Measurig Methods of distace betwee IFSs: - For two Ituitioistic fuzzy sets of X are deoted by, with truth-membership t A, t B ad false-membership f A, f B, respectively. Ataassove suggested the distace as follows: The Hammig distace l ˆ( AB, ) is give by ISSN : 9-3345 Vol. 4 No. 07 Jul 03 99
Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) lˆ( A, B ) = ( ta ( xi ) tb ( xi ) + fa ( xi ) fb ( xi )) () i = The Euclidea distace qˆ ( AB, ) is give by qˆ( A, B) = ( ta( xi) tb( xi) + fa( xi) fb( xi) ) ( i = Szmidt ad Kacprzyk gave a geometrical iterprepatia of IFSs, ad the they proposed correspodig modified distaces uio took accout the three parameters of Ituitioistic fuzzy sets. The defiitios of distace give by them are as follows: The Hammig distace l ( AB, ) l ( A, B) = ( ta( xi) tb( xi) + fa( xi) fb( xi) + πa( xi) πb( xi)) (3) The Euclidea distace q ( AB, ) is give by q ( A, B) = ( ta( xi) tb( xi) + fa( xi) fb( xi) + πa( xi) πb( xi) ) (4 Based o Hausdroff metric, Grzegorzewski proposed aother group of distaces. The Hammig distace l ( AB, ) is give l ( A, B) = max( t ( x ) t ( x ), f ( x ) f ( x )) (5) A i B i A i B i i = The Euclidea distace q ( AB, ) is give by q ( A, B) = max( t ( x ) t ( x ), f ( x ) f ( x ) ) (6 A i B i A i B i i = Lu ad Wag proposed a distace measuremet d sa( xi) sb( xi) ta( xi) tb( xi) fa( xi) fb( xi) ( AB, ) = ( + + ) (7) = 4 4 4 Where i s ( x ) = t ( x ) f ( x ) A i A i A i s ( x ) = t ( x ) f ( x ) B i B i B i New Distace betwee IFSs:- As we kow, the distace suggested by Ataassove[,] are the orthogoal projectios of the distace preseted by Szmidt[0] ad Kacprzyk. I this preset paper, we first correlate the distace suggested by Ataassove[,] ad Grzegorzewski,[4] ad the propose a ew group of distace to evaluate the fuzzy reliability of the system. Besides the Hammig ad Euclidea distace, for two ordiary fuzzy sets A, B of X, with membership fuctio t A, t B, the ormalized Mikowski s distace ca be defied as follows: l m / p p m ta( xi) tb( xi) ( AB, ) = (8) Naturally, we wat to exted it to IFS, p p m ta( xi) tb( xi) + fa( xi) fb( xi) l ( AB, ) = (9) ISSN : 9-3345 Vol. 4 No. 07 Jul 03 90
Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) It is easy to verify that equatio (9) will geerate to equatio () ad () whe p=, respectively. 3. Model Descriptio:- This model shows a device desiged to heavy weights. Three steel ropes are coected to two heavy ed plates J ad K. Each ed plate has two U-liks bolted to it ad each U-lik is held by four bolts. Whe the applied load exceeds the desig load, failure of the device may occur due to oe or more of the followig causes: (a) Failure of the bolts holdig each U-lik, sice there are four bolts holdig each lik, the failure of each of there bolts may be deoted by R, R, R 3,R 4. (b) Ay two of the steel ropes, or all the three ropes may fail due to over stressig. Lets R deotes the failure of two ropes ad R 3 deotes the failure of all the three ropes. (c) The fixtures of the ropes to each of the ed plates may fail. Deote these by C ad C referrig to the left ad right ed plate fixtures respectively. ISSN : 9-3345 Vol. 4 No. 07 Jul 03 9
Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) Device Rope fixtures U - liked Rope Over stregsed Rope fixtures of J Rope fixtures of K Two ropes Three ropes U- liks at ed J U- liks at ed K AND AND U - lik L U - lik L U - lik L 3 U - lik L 3 U - lik L 4 ISSN : 9-3345 Vol. 4 No. 07 Jul 03 9
Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) 4. Numerical Computatios:- Accordig to arithmetic operatios of triagle Ituitioistic fuzzy sets the failure rage of system failure show i the model ca be described as: = [{.. } { } ( )( )( ) ( D )( )( ) D E E F F ( G )( G )( G ) T A A B B C Z {... } { } 3 = {(-) [(0.99, 0.995, 0.998); 0.8], [(0.99, 0.995,.0); 0.9] ( ) [(0.99, 0.993, 0.994); 0.8], [(0.99, 0.993, 0.997); 0.9] ( ) [(,, ); 0.6], [(,, ); 0.7] ( ) [(0.99, 0.99, 0.993); 0.7], [(0.99, 0.99, 0.994); 0.8] ( ) [(,, ); 0.6], [(,, ); 0.8] ( ) [(0.99974, 0.9999, 0.99995); 0.8], [(0.9997, 0.9999, 0.99999); 0.9] ( ) [(0.99, 0.994, 0.995); 0.8], [(0.99, 0.994, 0.997); 0.8] ( ) [(0.99, 0.993, 0.993);.0], [(0.99, 0.993, 0.993);.0] ( ) [(0.994, 0.995, 0.996); 0.8], [(0.993, 0.995, 0.997); 0.9]} = {(-) [(0.9570, 0.9650, 0.96934); 0.6], [(0.94596, 0.9650, 0.9788); 0.7]} = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.7] O the basis of above calculatios, we fid that the failure iterval of "ower failure system as followig = [(0.0466, 0.03750, 0.04030); 0.6], [(0.038, 0.03750, 0.04404); 0.7] Thus the reliability iterval of "ower failure system" ca be described as the followig vague umber. [(0.9540, 0.9650, 0.96934); 0.6], [(0.9596, 0.9650, 0.9788); 0.7] (.5.) Expressio (.5.) iterprets the reliability to lie i the iterval (0.9540, 0.96934) with truth value 0.6 ad i the iterval (0.9596, 0.9788) with truth value 0.7. It ca be observed that the crisp value of traditioal reliability lies withi the obtaied itervals. I order to fid the vague importace idex, we calculate T as the followigs: TC = [(0.0306, 0.0374, 0.04705); 0.6], [(0.08, 0.0374, 0.05376); 0.7] T A = [(0.087, 0.0367, 0.0396); 0.6], [(0.08, 0.0367, 0.04449); 0.7] T A = [(0.0579, 0.0369, 0.0396)0.6], [(0.0888, 0.0369, 0.04449); 0.7] T B = [(0.0383, 0.0307, 0.03865); 0.6], [(0.0493, 0.0307, 0.04545); 0.7] T B = [(0.0677, 0.0367, 0.0455); 0.6], [(0.0888, 0.0367, 0.04737); 0.7] T D = [(0.048, 0.0307, 0.0396); 0.6], [(0.0888, 0.0307, 0.04545); 0.7] T D = [(0.0383, 0.0974, 0.03865); 0.6], [(0.059, 0.0974, 0.04449); 0.7] T E = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.8] T E = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.8] T F = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.8] T F = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.7] T G = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.7] T G = [(0.03066, 0.03750, 0.04730); 0.6], [(0.08, 0.03750, 0.05404); 0.7] ISSN : 9-3345 Vol. 4 No. 07 Jul 03 93
Neeraj Lata / Iteratioal Joural of Computer Sciece & Egieerig Techology (IJCSET) T G3 = [(0.0066, 0.0750, 0.03730); 0.6], [(0.08, 0.0750, 0.04404); 0.7] The I.F.I. of all basic evets are calculated as followig: I ( T, TA ) = 0.00068, I ( T, TA ) = 0.04, I ( T, TC ) = 0.03085, I ( T, TZ ) = 0 I ( T, TB ) = 0.03774, I ( T, TB ) = 0.0408, I ( T, TD ) = 0.0384, I ( T, TD ) = 0.03869, I ( T, TE ) = 0, I ( T, TE ) = 0, I ( T, TF ) = 0, I ( T, TF ) = 0, I ( T, TG ) = 0 I ( T, TG ) = 0.00034, I ( T, TG3 ) = 0.084. Basic evets havig vague importace idex zero or a very small umber idicate that those evets play either o role or very egligible role i the top evet. These evets ca therefore be igored while calculatig the crisp reliability usig traditioal method. I the preset example, the fault tree give i Figure 3. Repeatig the calculatios of Sectio 3 after igorig above evets, oe gets the crisp reliability estimate of the Model Failure System to be 0.9899. Thus our approach of Fuzzy importace idex could be useful i avoidig the uderestimatio of the reliability of the system. 5.Coclusio A ew Ituitioistic fault tree aalysis model is proposed i this paper that modifies the fuzzy set arithmetic operatios for implemetig fault tree aalysis. roposed method leads to two iterval estimates of reliability with differet truth values. The reliability estimate obtaied by traditioal approach lies iside the itervals. This work also itroduces the cocept of Ituitioistic Fuzzy Importace Idex that helps i discardig uimportat evets from the classical fault tree aalysis to avoid uder/ over estimatio of reliability. Results of Ituitioistic fault tree aalysis are more flexible tha the fuzzy fault tree aalysis because the later method caot describe the ucertaity of cofidece level. Refereces: [] K.T. Ataassov, Ituitioistic Fuzzy Sets, hysica-verlag, Heidelberg, f J/Kevi York, 999. [] K.T. Ataassov, More o ituitioistic fuzzy sets, Fuzzy Sets ad Systems 33() (989) 37-46. [3] L.A. Zadeh, Fuzzy sets. Iformatio Cotrol 8 (965) 338-353. [4] Lee J.H., Lee-Kwag H., Lee K.M, "A method for rakig fuzzily fuzzy umbers," 000 IEEE Iteratioal Fuzzy Systems Coferece, vol., pp.7-76, SaAtoio,Texas, May. 000. [5] Lee Youg-Hyu, Koop R., Applicatio of fuzzy cotrol for a hydraulic forgig machie. Fuzzy Sets ad Systems 8 (00) 99-08. [6] Lee-Kwag H., Lee J.H., "A method for rakig fuzzy umbers based o a viewpoit ad its applicatio to decisio makig," IEEE Tras. o Fuzzy Systems, vol.7, pp.677-685, Dec. 999. [7] Liao, H. ad Elsayed, E. A., Reliability redictio ad Testig la Based o a Accelerated Degradatio Rate Model, Iteratioal Joural of Materials & roduct Techology, Vol., No. 5, 40-4, 004 [8] S.M. Che, "Measures of similarity betwee vague sets." Fuzzy Sets & Systems 74(995) 7-3. [9] Sgarro Adrea, ossibilistic iformatio theory: a codig theoretic approach Fuzzy Sets ad Systems 3 (00) 3 [0] Siger D., Fault tree aalysis based o fuzzy logic, Computer Chem. Egg. Vol. 4, No3 pp59-66, 990 [] Che Je-Yag, Rule regulatio of fuzzy slidig mode cotroller desig: direct adaptive approach, Fuzzy Sets ad Systems 0(00) 59-68. [] Che Shyi- Mig, Aalyzig Fuzzy System Reliability Usig Vague Set Theory, Iteratioal joural of Applied Sciece ad Egieerig 003.,: 8-88. [3] Chea Y.H., Wagb We-Jue, Chih-Hui Chiub New estimatio method for the membership values i fuzzy sets, Fuzzy Sets ad Systems (000) {55} [4] Grzegorzewski,. ''Distaces betwee ituitioistic fuzzy sets ad /or iterval-valued fuzzy sets based o (ho Hausdorff metric", Fuzzy Sets ad Systems, Vol 48, pp. 3 9-38, 004. G [5] Szmidt, E, Kacprzyk. J. "Ituitioistic fuzzy sets i decisio makig", Notes IS, Vol. No.. pp. 5-3. 996. ISSN : 9-3345 Vol. 4 No. 07 Jul 03 94