Fuzzy Reliability of a Marine Power Plant Using Interval Valued Vague Sets

Transcription

1 Iteratoal Joural of Appled Scece ad Egeerg : 7-8 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets Amt Kumar a Shv Prasad Yadav a * ad Suredra Kumar b a Departmet of Mathematcs Ida Isttute of Techology Roorkee Roorkee Ida b Departmet of Electrcal Egeerg Ida Isttute of Techology Roorkee Roorkee Ida Abstract: I covetoal relablty aalyss the falure probabltes of the compoets of a system are treated as exact values. It s ofte dffcult to obta data for falure probabltes uder chagg evrometal codtos. Hece fuzzy sets are used to aalyze the fuzzy system relablty a fuzzy umber represets the relablty of each compoet. Che [4] aalyzed the fuzzy system relablty usg vague set theory. The values of the membershp ad o-membershp of a elemet a vague set are represeted by a real umber [0 ]. A specalst s always ucerta about the values of the membershp ad o-membershp of a elemet a set. Hece t s better to represet the values of the membershp ad o-membershp of a elemet a set by tervals of possble real umbers stead of real umbers. I ths paper a ew method has bee developed for aalyzg the fuzzy system relablty of a seres ad parallel system usg terval valued trapezodal vague sets the relablty of each compoet of each system s represeted by a terval valued trapezodal vague set defed the uverse of dscourse [0 ].The developed method has bee used to aalyze the fuzzy relablty of a mare power plat. The major advatage of the proposed approach (the cocept of terval valued vague sets over the exstg approaches [7 8 4] s that the proposed approach separates the postve ad egatve evdece for the membershp of a elemet a set. Also the proposed approach the values of the membershp ad o- membershp of a elemet a set are tervals stead of a sgle real umber. Keywords: fuzzy system relablty; vague set theory; fuzzy fault tree; terval valued vague sets.. Itroducto Oe of the mportat egeerg tasks desg ad developmet of a techcal system s relable egeerg. It s well kow that the covetoal relablty aalyss usg the probabltes has bee foud to be adequate to hadle ucertaty of falure data ad modelg. To overcome ths problem the cocept of fuzzy approach has bee used the evaluato of the relablty of a system. uzzy set theory was frst troduced by Zadeh [] 965. Sger [] preseted a fuzzy set approach for fault tree ad relablty aalyss. Ca et al. [3 4 5] gave a dfferet sght by troducg the possblty assumpto ad fuzzy state assumpto to replace the probablty ad bary state assumptos. * Correspodg author; e-mal:yadavfma@tr.eret. Accepted for Publcato: October Chaoyag Uversty of Techology ISSN It. J. Appl. Sc. Eg

2 Amt Kumar Shv Prasad Yadav ad Suredra Kumar Mo et al. [7] preseted a method for the fuzzy system relablty aalyss for compoets wth dfferet membershp fuctos va o-lear programmg techques. Che [8] preseted a method for fuzzy system relablty aalyss usg fuzzy umber arthmetc operatos. The collecto of papers by Osawa ad Kacprzyk [9] presets may other dfferet approaches for the fuzzy relablty. Ca [] preseted a troducto to system falure egeerg. Che [4] preseted a ew method for aalyzg the fuzzy system relablty based o vague sets. I ths paper the cocept of vague sets s exteded by dea of terval valued vague sets. Also we have troduced: some deftos related to terval valued vague set defto of terval valued trapezodal vague set ad arthmetc operatos betwee two terval valued trapezodal vague sets. urther a ew method has bee developed for aalyzg the fuzzy system relablty of a seres ad parallel system usg terval valued trapezodal vague sets the relablty of each compoet of each system s represeted by a terval valued trapezodal vague set defed the uverse of dscourse [0].The developed method has bee used to aalyze the fuzzy relablty of a mare power plat. The proposed method ca model ad aalyze the fuzzy system relablty a more flexble ad tellget maer comparso to the method gve by Che [4]. Ths paper s orgazed as follows. Sectos presets some deftos. Secto 3 presets the arthmetc operatos betwee two terval valued trapezodal vague sets. Secto 4 presets the fuzzy relablty calculatos of a seres ad parallel system usg terval valued trapezodal vague sets. Secto 5 presets the case study of the system. Secto 6 presets the algorthm for aalyzg fuzzy relablty. Secto 7 presets the assumed data. Sectos 8 presets the results.the coclusos are dscussed secto 9.. Some deftos I ths secto some deftos are revewed ad some other deftos are troduced... Deftos revewed We brefly revew some deftos [6 4] here... Defto A terval valued fuzzy set (over a basc set X s specfed by a fucto T : X D([0] D([0] s the set of all tervals wth [0 ].e. for all x X T ( x s a terval [ ] 0... Defto t A vague set a basc set X s characterzed by a truth membershp fucto t : X [0] ad a false membershp fucto f f : X [0 ]. If the ge- x the erc elemet of X s deoted by the lower boud o the membershp grade of x derved from evdece for x s deoted by t ( ad the lower boud o the ega- x to of f ( x x s deoted by f ( t ( x ad x both assocate a real umber the terval [0] wth each pot X t ( x f ( x. A vague set the uverse of dscourse X s show g.. Whe X s cotuous a vague set ca be wrtte as t ( x f ( x / x x X X Whe X s dscrete a vague set ca be wrtte as t ( x f ( x / x x X. 7 It. J. Appl. Sc. Eg

3 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets t ( X f ( X f ( X f ( x t ( x t ( X 0 x gure. A vague set. X..3 Defto Let A = [a a ] ad B = [b b ] be two arbtrary tervals the the mmum of A ad B s represeted by MIN [A B] ad s defed by MIN ([a a ]; [b b ] = [m (a b m (a b ]..4 Defto The complemet of a terval A = [a a ] s deoted by A ad s defed by A = [-a -a ]... Deftos troduced The defto of terval valued vague set ad deftos related to terval valued vague set are troduced here... Defto A terval valued vague set over a basc set X s defed as a object of the form [ x ; T ( x ; ( x ] x X T : X D[0] ad : X D[0] are caled Truth membershp fucto ad alse membershp fucto respectvely ad D([0] s the set of all tervals wth [0 ]... Defto A terval valued vague set s sad to be covex f ad oly f x ad x X T x x M T x T x x M x ( x x [0 ]..3 Defto A terval valued vague umber s a terval valued vague subset the uverse of dscourse X that s both covex ad ormal...4 Defto A terval valued trapezodal vague set (over a basc set X ca be represeted by ( x y z w ; [ ]; [ ] as show g. 0 It. J. Appl. Sc. Eg

4 Amt Kumar Shv Prasad Yadav ad Suredra Kumar gure. Iterval valued trapezodal vague set. 3. Arthmetc operatos betwee two terval valued trapezodal vague sets Certa arthmetc operatos betwee two terval valued trapezodal vague sets are developed here. Let us cosder two terval valued trapezodal vague sets ad as show g. 3 T ( X ( X T T ( X ( X ( X ( X ( X 0 x y z w x y z w gure 3. Iterval valued trapezodal vague sets ad. T ( X 0 x y z w X X defed by ( x y z w ;[ ];[ ] ( x y z w ;[ ];[ ] 0 The arthmetc operatos betwee ad are defed as follows ( x x y y z z w w ; [m( m( ]; [m( m( ] ( x w y z z y w x ; ( Θ [m( m( ]; > [m( m( ] ( x x y y z z w w ; [m( m( ]; [m( m( ] ( x / w y / z z / y w / x ; [m( m( ]; [m( m( ] 4. uzzy relablty calculatos of a seres ad parallel system usg terval valued trapezodal vague sets I ths secto a ew method has bee developed for the fuzzy relablty calculato of a seres ad parallel system usg terval valued trapezodal vague sets the relablty of each compoet of each system s represeted by a terval valued trapezodal vague set defed o the uverse of dscourse [0 ]. 4.. Seres system Let us cosder a seres system cosstg of compoets as show g.4. The fuzzy relablty R of the seres system show below ca be evaluated as follows: 74 It. J. Appl. Sc. Eg

5 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets R R gure 4. Seres system. [( x y z w ; [m( m( ];[m( m( ]] R R R [( x y z w ;[ ];[ ] represets the relablty of the th compoet. 4.. Parallel system R 5. Case Study A mare power plat [3] has two geerators G ad G oe located at the ster ad the other at bow. Each geerator s coected to ts respectve mcro swtch board- ad mcro swtch board-. The dstrbutve swtch board receves the supply from the swtch boards through cables C ad C ad respectve jucto boxes D ad E.The two mcro swtch boards are tercoected through a log cable C 3 ad the jucto boxes A ad B. The schematc dagram s show g. 6. R R R gure 5. Parallel system. Let us cosder a parallel system cosstg of compoets as show g.5. The fuzzy relablty R of the parallel system ca be evaluated as follows: + G G + R Θ ( Θ R [( ( x ( y ( z ( w ; [m( m( ]; [m( ' m( ' ]] MSB C 3 D A B E C DSB gure 6. Mare power plat. MSB Let us assume that basc compoets subjected to falure are (a Geerators G ad G. (b Mcro swtch board- (MSB- ad Mcro C It. J. Appl. Sc. Eg

6 Amt Kumar Shv Prasad Yadav ad Suredra Kumar swtch board- (MSB-. (c Itercoectg cable C 3 ad jucto boxes A ad B all are treated as oe ut. (d Jucto boxes D ad E. (e Dstrbutve swtch board (DSB. 5. Notatos x y z t ;[ ];[ ' ' ] represets the urelablty of geerator G. x y z t ;[ ];[ ' ' ] represets the urelablty of geerator G. x y z t ;[ ];[ ' ' ] represets the urelablty of mcro swtch board-. 4 x4 y4 z4 t4 ;[ 4 4 ];[ ' 4 ' 4 ] represets the urelablty of mcro swtch board-. x y z t ;[ ];[ ' ' ] represets the urelablty of the jucto boxes A ad B. x y z t ;[ ];[ ' ' ] represets the urelablty of the jucto box D. x y z t ;[ ];[ ' ' ] represets urelablty of the jucto box E. x y z t ;[ ];[ ' ' ] represets the urelablty of the dstrbutve swtch board. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply through the jucto boxes A ad B. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply to mcro swtch board-. X ;[ Y Z T ];[ ' ' ] represets the relablty of the evet that there s o power supply to the jucto box D. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply from the jucto box D. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply through the jucto boxes D ad E. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply to mcro swtch board-. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply to the jucto box E. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that there s o power supply from the jucto box E. X Y Z T ;[ ];[ ' ' ] represets the relablty of the evet that o power s comg to dstrbutve swtch board. represets the fuzzy urelablty of the system. R represets the fuzzy relablty of the system. 5.. ault tree The fault tree for the mare power plat (g. 6 s show g It. J. Appl. Sc. Eg

7 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets Total alure of the System D.S.B s burt No power comg to D.S.B No power from jucto box D No power from jucto box E D s burt No power to jucto box D E s burt No power to jucto box E M.S.B- s burt No power to M.S.B- M.S.B- s burt No power to M.S.B- G s burt G s burt No supply through A ad B No supply through A ad B G s burt M.S.B- s burt G s burt M.S.B- s burt A ad B damaged A ad B damaged gure 7. ault tree of mare power plat. It. J. Appl. Sc. Eg

8 Amt Kumar Shv Prasad Yadav ad Suredra Kumar 5.3. uzzy fault tree Usg fault tree the fuzzy fault tree (g. 8 s costructed for evaluatg the fuzzy relablty of the above mare power plat. It s based o fuzzy logc ad ca hadle fuzzy arthmetc wth the applcato of fuzzy set theory. 6. Proposed algorthm Wth the help of fuzzy fault tree the followg algorthm s proposed for aalyzg the fuzzy relablty of the above mare power plat. Step Θ(Θ (Θ 5 (Θ 4 X Y Z T ;[ ];[ ' ' ] X ( x ( x ( x 5 4 Y ( y ( y ( y 5 4 Z ( z ( z ( z 5 4 T ( t ( t ( t 5 4 m( 4 5 m( 4 5 ' m( 4 5 ' ' ' ' m( ' ' ' 4 5 Step X Y Z T ;[ ];[ ' ' ] X x X Y y Y Z z Z T t T m ( 4 5 m ( 4 5 ' m ( ' ' ' ' 4 5 ' m ( ' ' ' ' Step Θ( Θ ( Θ 3 3 X 3 Y3 Z 3 T3 ;[ 3 3 ];[ ' 3 ' 3] X ( x ( X Y ( y ( Y Z ( z ( Z T ( t ( T m ( m ( ' m ( ' ' ' ' ' m ( ' Step 4 ' ' ' ' 4 Θ(Θ 6 (Θ 3 ( X 4 Y4 Z4 T4;[ 4 4 ];[ ' 4 ' 4] X ( x ( X Y ( y ( Y Z ( z ( Z T ( t ( T m( m( ' 4 m ( ' ' ' ' ' ' ' m( ' ' ' ' ' ' Step 5 5 =Θ(Θ (Θ 5 (Θ 3 78 It. J. Appl. Sc. Eg

9 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets X ( x ( x ( x Y ( y ( y ( y Z ( z ( z ( z T ( t ( t ( t ( X Y Z T ;[ ];[ ' ' ] m ( m ( ' m ( ' ' ' ' m ( ' ' ' Step X6 Y6 Z6 T6 ;[ 6 6 ];[ ' 6 ' 6] X x X Y y Y Z z Z T t T m ( m( ' m ( ' ' ' ' ' m ( ' ' ' ' Step Θ(Θ ( Θ X 7 Y7 Z 7 T7 ;[ 7 7 ];[ ' 7 ' 7] X ( x ( X Y ( y ( Y Z ( z ( Z T ( t ( T m( m( ' m( ' ' ' ' ' ' m( ' ' ' ' ' OR OR OR OR OR 5 3 gure 8. uzzy fault tree. Step 8 8 Θ(Θ 7 ( Θ 7 X 8 Y8 Z 8 T8 ;[ 8 8 ];[ ' 8 ' 8] OR 4 3 AND AND 9 AND OR OR 5 It. J. Appl. Sc. Eg

10 Amt Kumar Shv Prasad Yadav ad Suredra Kumar X ( x ( X Y ( y ( Y Z ( z ( Z T ( t ( T m( m( ' m( ' ' ' ' ' ' ' m( ' ' ' ' ' ' Step X9 Y9 Z9 T9 ;[ 9 9 ];[ ' 9 ' 9] X X X Y Y Y Z Z Z T T T m( m( ' m( ' ' ' ' ' ' ' ' m( ' ' ' ' ' ' ' Step 0 Θ(Θ 8 (Θ 9 ( x 8( X 9 ( y 8( Y 9 ( z ( Z ( t ( T ; [m( ; m( ]; [m( ' ' ' ' 3 4 ' ' ' ' ; m ( ' ' ' ' 7. Data ' ' ' ' ] The followg data terms of terval valued trapezodal vague sets are assumed for aalyzg the fuzzy relablty of the above mare power plat: < [( ; [ ]; [ ]]> < [( ; [ ]; [ ]]> 3 < [( ; [ ]; [ ]]> 4 < [( ; [ ]; [ ]]> 5 < [( ; [ ]; [ ]]> 6 < [( ; [ ]; [ ]]> 7 < [( ; [ ]; [ ]]> 8 < [( ; [ ]; [ ]]> 8. Results The fuzzy urelablty ad the fuzzy relablty R Θ of the above mare power plat has bee computed usg the above data ad the proposed algorthm ad obtaed as the followg terval valued trapezodal vague sets < [( ; [ ]; [ ]]> ad R < [( ; [ ]; [ ]]> Iterval valued trapezodal vague sets represetg ad R are show gs. 9 ad 0 respectvely. 80 It. J. Appl. Sc. Eg

11 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets T ( r ( r Urelablty ( r Les from top to bottom: Represets supremum value of the terval ( r for each r r Represets fmum value of the terval ( r for each r r sets ad vague sets s that terval valued vague sets separate the postve ad egatve evdece for the membershp of a elemet a set ad also the values of membershp ad o membershp of a elemet a set are tervals stead of a sgle real umber. Iterval valued trapezodal vague sets are used whch s very effectve represetg the falure possblty of the basc evets uder fuzzy evromet. T R ( r R ( r gure 9. A terval valued trapezodal vague set represetg. 9. Cocluso Represets supremum value of the terval T ( for each r r r Represets fmum value of the terval T ( for each r r r I ths paper a ew method has bee developed for aalyzg the fuzzy system relablty of a seres ad parallel system usg terval valued trapezodal vague sets the relablty of each compoet of each system s represeted by a terval valued trapezodal vague set defed the uverse of dscourse [0 ]. The developed method has bee used to aalyze the fuzzy relablty of a mare power plat. The major advatage of usg terval valued vague sets over fuzzy Relablty ( r Les from top to bottom: Represets supremum value of the terval ( r R for each r r Represets fmum value of the terval ( r R for each r r Represets supremum value of the terval T ( for each r r R r Represets fmum value of the terval T R ( r for each r r gure 0. A terval valued trapezodal vague set represetg R. It. J. Appl. Sc. Eg

12 Amt Kumar Shv Prasad Yadav ad Suredra Kumar Ackowledgemet The authors are thakful to the revewers for ther crtcal ad frutful commets. The authors also ackowledge the facal support gve by the Coucl of Scetfc ad Idustral Research Govt. of Ida INDIA Refereces [ ] Zadeh L. A uzzy sets. Iformato Cotrol; 8: [ ] Sger D A fuzzy set approach to fault tree ad relablty aalyss. uzzy Sets ad Systems 34 : [ 3] Ca K.Y. We C.Y. ad Zhag M.L. 99. uzzy varables as a bass for a theory of fuzzy relablty the possblty cotext. uzzy Sets ad Systems 4 : [ 4] Ca K.Y. We C.Y. ad Zhag M.L. 99. Posbst relablty behavor of typcal systems wth two types of falures. uzzy Sets ad Systems 43 :7-3. [ 5] Ca K.Y. We C.Y. ad Zhag M.L uzzy states as a bass for a theory of fuzzy relablty. Mcroelectroc Relablty 33 : [ 6] Gau W.L. ad Buehrer D.J ague sets. IEEE.Tras.System Ma Cyber 3 : [ 7] Mo D.L. ad Cheg C.H uzzy system relablty aalyss for compoets wth dfferet membershp fuctos. uzzy Sets ad Systems 64 : [ 8] Che S.M.994. uzzy system relablty aalyss usg fuzzy umber arthmetc operatos. uzzy Sets ad Systems 64 :3-38. [ 9] Osawa T. ad Kacprzyk J Relablty ad safety uder fuzzess. st ed.: Physca erlag. [0] Che S.M Arthmetc operatos betwee vague sets. Proc. of the It. Jot Coferece of CSA/IIS/SO T o uzzy theory ad Applcatos 06-. [] Ca K.Y System falure egeerg ad fuzzy methodology: a troductory overvew. uzzy Sets ad Systems 83 : [] Zmmerma H.J uzzy set theory ad ts applcato. d ed.: Alled Publsher Ltd. [3]Srath L.S Relablty egeerg. 3 rd ed.: East-west Press Pvt. Ltd. [4] Che S.M Aalyzg fuzzy system relablty usg vague set theory. Iteratoal Joural of Appled Scece ad Egeerg : It. J. Appl. Sc. Eg