Analyzing Fuzzy System Reliability Using Vague Set Theory

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1 Iteratoal Joural of Appled Scece ad Egeerg 2003., : Aalyzg Fuzzy System Relablty sg Vague Set Theory Shy-Mg Che Departmet of Computer Scece ad Iformato Egeerg, Natoal Tawa versty of Scece ad Techology, Tape 06, Tawa, R. O. C. Abstract: Relablty modelg s the most mportat dscple of relable egeerg. A ew method for aalyzg fuzzy system relablty usg vague set theory s demostrated, where the relabltes of the compoets of a system are represeted by vague sets defed the uverse of dscourse [0, ]. The proposed method ca model ad aalyze the fuzzy system relablty a more flexble ad more tellget maer. Keywords: fuzzy system relablty; false-membershp fucto; truth-membershp fucto; vague sets.. Itroducto I [8], Kaufma et al. poted out that relablty modelg s the most mportat oe of the dscples of relable egeerg. The covetoal relablty of a system behavor s fully characterzed the cotext of probablty measures. However, because of the accuracy ad ucertates of data, the estmato of precse values of probablty becomes very dffcult may systems []. I recet years, some researchers focused o usg fuzzy set theory [23] for fuzzy system relablty aalyss [2-5], [7], [8], [], [2], [5], [9], [20]. I [3], Ca et al. preseted the followg two assumptos: () Fuzzy-state assumpto: At ay tme, the system may be ether the fuzzy success state or the fuzzy falure state. (2) Possblty assumpto: The system behavor ca be fully characterzed by possblty measures. I [5], Ca preseted a troducto to system falure egeerg ad ts use of fuzzy methodology. I [7], we preseted a method for fuzzy system relablty aalyss usg fuzzy umber arthmetc operatos. I [], we preseted a method for fuzzy system relablty aalyss based o fuzzy tme seres ad theα cuts operatos of fuzzy umbers. I [3], Cheg et al. preseted a method for fuzzy system relablty aalyss by terval of cofdece. I [9], Mo et al. preseted a method for fuzzy system relablty aalyss for compoets wth dfferet membershp fuctos va o-lear programmg techques. I [20], Sger preseted a fuzzy set approach for fault tree ad relablty aalyss. I [2], Suresh et al. preseted a comparatve study of probablstc ad fuzzy methodologes for ucertaty aalyss usg fault trees. I [22], tk et al. preseted a system of fuctoal equatos for fuzzy relablty aalyss of varous systems the possblty cotext. * Correspodg author: E-mal: smche@et.tust.edu.tw Accepted for Publcato: Feb. 26, Chaoyag versty of Techology, ISSN It. J. Appl. Sc. Eg., 2003.,

2 Aalyzg Fuzzy System Relablty sgvague Set Theory I ths paper, we preset a ew method for aalyzg fuzzy system relablty usg vague sets [0], [6], where the relabltes of the compoets of a system are represeted byvague sets defed the uverse of dscourse [0, ]. The grade of membershp of a elemet x a vague set s represeted by a vague value [t x, f x ] [0, ], where t x dcates the degree of truth, f x dcates the degree of false, t x f x dcates the ukow part, 0 t x f x, ad t x + f x. The oto of vague sets s smlar to that of tutostc fuzzy sets [], ad both of them are geeralzatos of the oto of fuzzy sets [23]. The proposed method ca model ad aalyze fuzzy system relablty a more flexble ad more tellget maer. It ca provde us wth a more flexble ad more tellget way for aalyzg fuzzy system relablty. Ths paper s orgazed as follows. I Secto 2, we brefly revew some deftos ad arthmetc operatos of vague sets from [0] ad [6]. I Secto 3, we preset a ew method for aalyzg fuzzy system relablty based o vague set theory. The coclusos are dscussed Secto Basc cocepts of vague sets I 965, Zadeh proposed the theory of fuzzy sets [23]. Roughly speakg, a fuzzy set s a class wth fuzzy boudares. Let be the uverse of dscourse, = {u, u 2,, u }. The grade of membershp of a elemet u a fuzzy set s represeted by a real value betwee zero ad oe, where u. I [6], Gau et al. poted out that ths sgle value combes the evdece for u ad the evdece agast u. It does ot dcate the evdece for u ad the evdece agast u, respectvely, ad t does ot dcate how much there s of each. Furthermore, Gau et al. also poted out that the sgle value tells us othg about ts accuracy. Thus, [6], Gau et al. preseted the cocepts of vague sets. I [0], we preseted the arthmetc operatos betwee vague sets. Let be the uverse of dscourse, = {u, u 2,, u }, wth a geerc elemet of deoted by u. A vague set the uverse of dscourse s characterzed by a truth-membershp fucto t, t : [0, ], ad a false-membershp fucto f, f : [0, ], where t (u ) s a lower boud of the grade of membershp of u derved from the evdece for u, f (u ) s a lower boud o the egato of u derved from the evdece agast u, ad t (u ) + f (u ). The grade of membershp of u the vague set s bouded by a subterval [t (u ), - f (u )] of [0, ]. The vague value [t (u ), - f (u )] dcates that the exact grade of membershp μ (u ) of u s bouded by t (u ) μ (u ) - f (u ), where t (u ) + f (u ). For example, a vague set the uverse of dscourse s show Fgure. t (), - f ().0 - ƒ (u ) t () t (u ) 0 u Whe the uverse of dscourse s a fte set, a vague set of the uverse of dscourse ca be represeted as = [t (u ),- f - f () Fgure. A vague set. (u )] / u. () It. J. Appl. Sc. Eg., 2003., 83

3 Shy-Mg Che Whe the uverse of dscourse s a fte set, a vague set of the uverse of dscourse ca be represeted as = [t (u ),- f (u )] / u, u. (2) Defto 2.: Let be a vague set of the uverse of dscourse wth the truthmembershp fucto t ad the falsemembershp fucto f, respectvely. The vague set s covex f ad oly f for all u, u 2, t (λu + (-λ)u 2 ) M(t (u ), t (u 2 )), (3) - f (λu + ( -λ) u 2 ) M( - f (u ), - f (u 2 )), (4) whereλ [0, ]. Defto 2.2: A vague set of the uverse of dscourse s called a ormal vague set f u, such that - f (u ) =. That s, f (u ) = 0. Defto 2.3: A vague umber s a vague subset the uverse of dscourse that s both covex ad ormal. I the followg, we troduce some arthmetc operatos of tragular vague sets [0]. Let us cosder the tragular vague set show Fgure 2, where the tragular vague set ca be parameterzed by a tuple <[(a, b, c); µ ], [(a, b, c);. For coveece, the tuple <[(a, b, c); µ ], [(a, b, c); ca also be abbrevated to <[(a, b, c); µ ;, where 0 µ µ 2. Some arthmetc operatos betwee tragular vague sets are as follows: Case : Cosder the tragular vague sets ad B ~ show Fgure 3, where = <[(a, b, c ); µ ], [(a, b, c ); = <[(a, b, c ); µ ;, B ~ = <[(a 2, b 2, c 2 ); µ ], [(a 2, b 2, c 2 ); = <[(a 2, b 2, c 2 ); µ ;, ad 0 µ µ 2. The arthmetc operatos betwee the tragular vague sets ad B ~ are defed as follows: B ~ = <[(a, b, c ); µ ], [(a, b, c ); <[(a 2, b 2, c 2 ); µ ], [(a 2, b 2, c 2 ); = <[(a + a 2, b + b 2, c + c 2 ); µ ], [(a + a 2, b + b 2, c + c 2 ); = <[(a + a 2, b + b 2, c + c 2 ); µ ;, (5) B ~ = <[(a 2, b 2, c 2 ); µ ], [(a 2, b 2, c 2 ); µ 2 ]> <[(a, b, c ); µ ], [(a, b, c ); = <[(a 2 - c, b 2 - b, c 2 - a ); µ ], [(a 2 - c, b 2 - b, c 2 - a ); = <[(a 2 - c, b 2 - b, c 2 - a ); µ ;, (6) B ~ = <[(a, b, c ); µ ], [(a, b, c ); µ 2 ]> <[(a 2, b 2, c 2 ); µ ], [(a 2, b 2, c 2 ); = <[(a a 2, b b 2, c c 2 ); µ ], [(a a 2, b b 2, c c 2 ); = <[(a a 2, b b 2, c c 2 ); µ ;, (7) B ~ = <[(a 2, b 2, c 2 ); µ ], [(a 2, b 2, c 2 ); <[(a, b, c ); µ ], [(a, b, c ); = <[(a 2 /c, b 2 /b, c 2 /a ); µ ], [(a 2 /c, b 2 /b, c 2 /a ); = <[(a 2 /c, b 2 /b, c 2 /a ); µ ;. (8) t (), - f () µ 2 - f () µ 0 t () a b c Fgure 2. A tragular vague set. 84 It. J. Appl. Sc. Eg., 2003.,

4 Aalyzg Fuzzy System Relablty sgvague Set Theory t (), - f () t (), - f B ~ () B ~ μ 2 μ - f () - f B ~ () t ~() () A t B ~ 0 a b c a 2 b 2 c 2 = <[(a a 2, b b 2, c c 2 ); M(µ, µ 3 )], [(a a 2, b b 2, c c 2 ); M(µ 2, µ 4 )]>, () B ~ = <[(a 2, b 2, c 2 ); µ 3 ], [(a 2, b 2, c 2 ); µ 4 ]> <[(a, b, c ); µ ], [(a, b, c ); = <[(a 2 /c, b 2 /b, c 2 /a ); M(µ, µ 3 )], [(a 2 /c,b 2 /b, c 2 /a ); M(µ 2, µ 4 )]>. (2) Fgure 3. Tragular vague sets ad B ~ (Case ). t (), - f () t (), - f B ~ () B ~ Case 2: Cosder the tragular vague sets ad B ~ show Fgure 4, where μ 2 - f () - f B ~ () A ~ = <[(a, b, c ); µ ], [(a, b, c );, μ 4 μ t () B ~ = <[(a 2, b 2, c 2 ); µ 3 ], [(a 2, b 2, c 2 ); µ 4 ]>, μ 3 t B ~ () ad 0 µ 3 µ µ 4 µ 2. The arthmetc operatos betwee the tragular vague sets ad B ~ are defed as follows: 0 a b c a 2 b 2 c 2 B ~ = <[(a, b, c ); µ ], [(a, b, c ); <[(a 2, b 2, c 2 ); µ 3 ], [(a 2, b 2, c 2 ); µ 4 ]> = <[(a + a 2, b + b 2, c + c 2 ); M(µ, µ 3 )], [(a + a 2, b + b 2, c + c 2 ); M(µ 2, µ 4 )]>, ( 9) B ~ = <[(a 2, b 2, c 2 ); µ 3 ], [(a 2, b 2, c 2 ); µ 4 ]> <[(a, b, c ); µ ], [(a, b, c ); = <[(a 2 - c, b 2 - b, c 2 - a ); M(µ, µ 3 )], [(a 2 - c, b 2 - b, c 2 - a ); M(µ 2, µ 4 )]>, (0) B ~ = <[(a, b, c ); μ ], [(a, b, c ); <[(a 2, b 2, c 2 ); µ 3 ], [(a 2, b 2, c 2 ); µ 4 ]> Fgure 4. Tragular vague sets ad B ~ (Case 2). 3. Aalyzg fuzzy system relablty based o vague sets I ths secto, we preset a ew method for aalyzg fuzzy system relablty based o vague set theory, where the relabltes of the compoets of a system are represeted by vague sets defed the uverse of dscourse [0, ]. Cosder a seral system show Fgure 5, where the relablty R ~ of compoet P s represeted by a vague set <[(a, b, c ); µ ;, where 0 µ µ 2, ad. The, the relablty R ~ of the seral system show Fgure 5 ca be evaluated as follows: R ~ = R ~ R ~ 2 R ~ It. J. Appl. Sc. Eg., 2003., 85

5 Shy-Mg Che = <[(a, b, c ); µ ; <[(a 2, b 2, c 2 ); µ 2 ; µ 22 ]> <[(a, b, c ); µ ; P = <[( a, b, c ); M(µ, µ 2,, µ ); M(µ 2, µ 22,, µ 2 )]> Iput P 2 Output (3) P Iput P P 2 P Output Fgure 6. Cofgurato of a parallel system. Fgure 5. Cofgurato of a seral system. Furthermore, cosder the parallel system show Fgure 6, where the relablty R ~ of compoet P s represeted by a vague set <[(a, b, c );μ ;μ 2 ]>, where 0 μ μ 2, ad. The, the relablty R ~ of the parallel system show Fgure 6 ca be evaluated as follows: R ~ = ( R ~ ) = ( <[(a, b, c ); µ ; ) ( <[(a 2, b 2, c 2 ); µ 2 ; µ 22 ]>) ( <[(a, b, c ); µ ; ) = <[(- c, - b, - a ); µ ; < [( - c 2, - b 2, - a 2 ) ; µ 2 ; µ 22 ]> <[( - c, - b, - a ); µ ; = <[( ( - c ), ( - b ), ( - a )); M(µ ; µ 2,, µ ); M(µ 2 ; µ 22,, µ 2 )]> = <[(- ( - a ), - ( - b ),- ( - c )); M(µ ; µ 2,, µ ); M(µ 2 ; µ 22,, µ 2 )]>. (4) I the followg, we use a example to llustrate the fuzzy system relablty aalyss process of the proposed method. Example 3.: Cosder the system show Fgure 7, where the relabltes of the compoets P, P 2, P 3 ad P 4 are R ~, R ~ 2, R ~ 3 ad R ~ 4, respectvely, where R ~ = <[(a, b, c ); µ ;, R ~ 2 = <[(a 2, b 2, c 2 ); µ 2 ; µ 22 ]>, R ~ 3 = <[(a 3, b 3, c 3 ); µ 3 ; µ 32 ]>, R ~ 4 = <[(a 4, b 4, c 4 ); µ 4 ; µ 42 ]>, 0 µ µ 2, ad 4. Based o the prevous dscusso, we ca see that the relablty R ~ of the system show Fgure 7 ca be evaluated as follows: R ~ = [ ( R ~ ) ( R ~ 2)] [ ( R ~ 3) ( R ~ 4)] = [ ( <[(a, b, c ); µ ; ) ( <[(a 2, b 2, c 2 ); µ 2 ; µ 22 ]>) [ ( <[(a 3, b 3, c 3 ); µ 3 ; µ 32 ]>) ( <[(a 4, b 4, c 4 ); µ 4 ; µ 42 ]>)] = [ <[( - c, - b, - a ); µ ; <[( - c 2, - b 2, - a 2 ); µ 2 ; 86 It. J. Appl. Sc. Eg., 2003.,

6 Aalyzg Fuzzy System Relablty sgvague Set Theory µ 22 ]>] [ <[( - c 3, - b 3, - a 3 ); µ 3 ; µ 32 ]> <[( - c 4, - b 4, - a 4 ); µ 4 ; µ 42 ]>] = [ <[(( - c )( - c 2 ), ( - b )( - b 2 ), ( - a )( - a 2 )); M(µ ; µ 2 ); M(µ 2, µ 22 )]>] [ <[(( - c 3 )( - c 4 ), ( - b 3 )( - b 4 ), ( - a 3 )( - a 4 )); M(µ 3 ; µ 4 ); M(µ 32, µ 42 )]>] = <[( - ( - a )( - a 2 ), - ( - b )( - b 2 ), - ( - c )( - c 2 )); M(µ ; µ 2 ); M(µ 2, µ 22 )]>] <[( - ( - a 3 )( - a 4 ), - ( - b 3 )( - b 4 ), - ( - c 3 )( - c 4 )); M(µ 3 ; µ 4 ); M(µ 32, µ 42 )]> = <[(a + a 2 - a a 2, b + b 2 - b b 2, c + c 2 - c c 2 ); M(µ ; µ 2 ); M(µ 2, µ 22 )]> <[(a 3 + a 4 - a 3 a 4, b 3 + b 4 - b 3 b 4, c 3 + c 4 - c 3 c 4 ); M(µ 3 ; µ 4 ); M(µ 32, µ 42 )]> = <[((a + a 2 - a a 2 )(a 3 + a 4 - a 3 a 4 ), (b + b 2 - b b 2 )(b 3 + b 4 - b 3 b 4 ), (c + c 2 - c c 2 )(c 3 + c 4 - c 3 c 4 )), M(µ, µ 2, µ 3, µ 4 ); M(µ 2, µ 22, µ 32, µ 42 )]> = <[(a a 3 + a a 4 - a a 3 a 4 + a 2 a 3 + a 2 a 4 - a 2 a 3 a 4 - a a 2 a 3 - a a 2 a 4 + a a 2 a 3 a 4, b b 3 + b b 4 - b b 3 b 4 + b 2 b 3 + b 2 b 4 - b 2 b 3 b 4 - b b 2 b 3 - b b 2 b 4 + b b 2 b 3 b 4, c c 3 + c c 4 - c c 3 c 4 + c 2 c 3 + c 2 c 4 - c 2 c 3 c 4 -c c 2 c 3 - c c 2 c 4 + c c 2 c 3 c 4 ); M(µ, µ 2, µ 3, µ 4 ); M(µ 2, µ 22, µ 32, µ 42 )]>. (5) 4. Coclusos We have preseted a ew method for aalyzg fuzzy system relablty usg vague set theory, where the compoets of a system are represeted by vague sets defed the uverse of dscourse [0, ]. The proposed method ca model ad aalyze fuzzy system relablty a more flexble ad more tellget maer. It ca provde us wth a more flexble ad more tellget way for fuzzy system relablty aalyss. Ackowledgemets Ths work was supported part by the Natoal Scece Coucl, Republc of Cha, uder Grat NSC E Refereces [ ] Ataassov, K Itutostc fuzzy sets. Fuzzy Sets ad Systems, 20: [ 2] Ca, K. Y., We, C. Y., ad Zhag, M. L. 99. Fuzzy varables as a bass for a theory of fuzzy relablty possblty cotext. Fuzzy Sets ad Systems, 42: [ 3] Ca, K. Y., We, C. Y., ad Zhag, M. L. 99. Posbst relablty behavor of typcal systems wth two types of falure. Fuzzy Sets ad Systems, 43: [ 4] Ca, K. Y., We, C. Y., ad Zhag, M. L. 99. Fuzzy relablty modelg of gracefully degradable computg systems. Relablty Egeerg ad System Safety, 33: [ 5] Ca, K. Y System falure egeerg ad fuzzy methodology: A troductory overvew. Fuzzy Sets ad Systems, 83: [ 6] Che, S. M A ew approach to hadlg fuzzy decsomakg problems. IEEE Trasactos o Systems, Ma, ad Cyberetcs, 8: [ 7] Che, S. M Fuzzy system relablty aalyss usg fuzzy umber arthmetc operatos. Fuzzy Sets ad Systems, 64: [ 8] Che, S. M. ad Jog, W. T Aalyzg fuzzy system relablty usg terval of cofdece. Iteratoal Joual of Iformato Maagemet ad Egeerg, 2: [ 9] Che, S. M New methods for subjectve metal workload assessmet ad It. J. Appl. Sc. Eg., 2003., 87

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