Calculation of Failure Probability of Series and Parallel Systems for Imprecise Probability

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1 IJ Egeerg ad Mauacturg, 2012,2, Publshed Ole Aprl 2012 MECS ( DOI: /jem Avalable ole at Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty B Suo a,*, Yog-sheg Cheg a,*, Chao Zeg a,*, Ju L a,* a Isttute o Electroc Egeerg, Cha Academy o Egeerg Physcs, Mayag, Provce, Cha Abstract I the stuato that ut alure probablty s mprecse whe calculato the alure probablty o system, classcal probablty method s ot applcable, ad the aalyss result o terval method s coarse To calculate the relablty o seres ad parallel systems above stuato, D-S evdece theory was used to represet the ut alure probablty Mult-sources ormato was used, ad bele ad plausblty ucto were used to calculate the relablty o seres ad parallel systems by evdetal reasog By ths mea, lower ad upper bouds o probablty dstrbuto o system alure probablty were obtaed Smulato result shows that the proposed method s preerable to deal wth the mprecse probablty relablty calculato, ad ca get addtoal ormato whe compare wth terval aalyss method Idex Terms: relablty; alure probablty; seres-parallel systems; D-S evdece theory 2012 Publshed by MECS Publsher Selecto ad/or peer revew uder resposblty o the Research Assocato o Moder Educato ad Computer Scece 1 Itroducto I relablty estmato o complex system, t s always the stuato that expermetal data s lmted, or ormato s ot completed As sample sze s small, ad expermet s ot eough, there are a varety o ucertates the system I ths case, t s hard to calculate the relablty o seres-parallel coecto uts Tradtoal methods o ucertaty represetato are probablty theory ad uzzy theory Both o them eed the probablty dstrbutos or membershp uctos o the ut alure probablty I realty, t s much more easer to get the tervals o ut alure probablty tha to get ther probablty dstrbutos Whe perormace ad relablty o system are evaluated ths stuato, the ut alure probablty s supposed to ollow certa dstrbutos artcally, the results o evaluato are always ot objectve, ad may lead to large aalyss errors [1] Except or probablty theory ad uzzy theory, there are a lot o methods to deal wth above ucertaty ormato, such as covex model, o-gap model, terval aalyss, ad D-S evdece theory * Correspodg author E-mal address: {suoyy, chys_ee, zch_bob & lju_ee}@163com

2 80 Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty Covex model s a o-probablty method, whch represet the ucertaty o parameters wth covex model, ad traslate the ucertaty o put parameters to the system respose qualty ad the problem target qualty by perormace optmzato ad robust optmzato [2][3] Io-gap model s developed based o covex model It s better tha other methods treatg wth small samples ad represetg epstemc ucertaty Io-gap model s cosstet wth covex model o ucertaty qualty represetato, but ts decso-makg approach s better [4-6] Iterval aalyss represets the ucertaty by the dstrbuto tervals o parameters, ad s very sutable to the stuato that the boudares o parameters are the oly valuable ormato [7-8] The solutos o terval equatos clude drect method, combed method, ad terval-trucato method But sometmes ther solutos are too pessmstc [9] I some sese, terval aalyss s a part o covex model, because the terval represetato o ut alure probablty s a specal example o covex sets [10] I D-S evdece theory, bele ucto s used to represet the precse bele degree o evdece or proposto, ad plausblty s used to represet the maxmum amout o lkelhood that the evdece s true As D-S evdece theory geeralze spot-value ucto to set ucto, ad ts basc research object s set or terval umber, t has great advatages represetg ad treatg wth ucertaty ormato [11-12] As dscussed prevous, probablty theory ad uzzy theory eed the dstrbutos ormato o ut alure probablty, whch s hard to obta realty Covex model ad o-gap model s avalable the stuato that the boudares o parameters are the oly useul ormato, ad ca draw a relatvely valuable cocluso wth lmted ormato But whe the boudares o ut alure probablty are kow, ad multple ormato sources about the probablty o ut alure probablty there sub-tervals are gve as well, these two methods are ot avalable The ormato sources may come rom deret expermets, tests ad smulatos, or come rom the hstorcal data o aalogous system, ad eve the judgmets o experts To calculate the relablty o seres-parallel systems above stuato, D-S evdece theory s used to represet the ut alure probablty Mult-sources ormato s used, ad bele ad plausblty ucto are used to calculate the relablty o seres-parallel systems by evdetal reasog 2 D-S Evdece Theory Deto 1: s the whole hypothess space, whch s a o-empty set I m s a mass ucto o, the ucto Bel : 2 [0,1] deed by [12] Bel( A) m( B) (1) B A s a bele ucto, ad the ucto Pl : 2 [0,1] deed by Pl( A) m( B) (2) BA s a plausblty ucto, where A 2 ad A They are related to each other by the ollowg equato Bel( A) 1 Pl( A) (3) Theorem 1: Let m1, m2,, m be the mass uctos o, ad the ocal elemets are A ( 1,2,, N), the

3 Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty K ma ( ) A A1 N m ( A) A 0 A = (4) where K A A1 N m ( A ) Eq (4) s the Dempster s rule o mult-sources ormato combato [13] Theorem 2: Let Bel ad Pl be the bele ucto ad plausblty ucto o respectvely, the Bel( A) Pl( A) (5) where A 2 [14] Eq (5) shows that as a measure o evet A s true, Pl( A ) s a more optmstc evaluato tha Bel( A ) I PA ( ) s the true value o the measure o set {A s true}, the Bel( A) P( A) Pl( A) (6) Theorem 3: Cosder a set A X, ad dee a mappg : A B, ad Y { B : B ( A), A X} s aother rame o dscermet, the Pl B Pl B (7) 1 Y( ) X( ( )) 1 Y( ) X( ( )) Bel B Bel B (8) The proo o (7) ad (8) ca be see Re [15] 3 Calculato o Falure Probablty o Seres ad parallel System Suppose each ut seres-parallel system s two-state evet, that s, workg or alure Every uts are depedet each other For a system o uts, alure probablty o th ut s expressed as a ucto wth several basc desg parameters F ( x, x,, x ), 1,2,, (9) 1 2 where x 1, x 2,, x s the basc desg parameters o th ut Falure probablty o system, P, s P g( F, F,, F ) (10) 1 2 where ucto g () descrbe the relatoshp o each uts For th ut, because o the ucertates o desg parameters X { x 1, x 2,, x}, F s mprecse I practce, the terval value o F could be obtaed by expermets, smulatos, ad estmates o experts Let F be the rame o dscermet o uts F { F1, F2,, F }, ad rame o dscermet o alure probablty o system s P { B : B g( A ), A F} For seres system, as show Fg 1, alure probablty o system s

4 82 Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty P g( F ) 1 (1 F ) (11) Fg 1 Seres system For parallel system,as show Fg 2, alure probablty o system s 1 P g( F ) F (12) 1 2 Fg 2 Parallel system For parallel-seres system, as show Fg 3, alure probablty o system s 2 4 P g( F ) 1 (1 F )(1 F ) (13) Fg 3 Parallel-seres system For seres-parallel system, as show Fg 4, alure probablty o system s 2 4 P g( F ) (1 (1 F ))(1 (1 F )) (14) Fg 4 Seres-parallel system From (7) ad (8), the

5 P(<p ) Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty 83 Bel ( p p) Bel ( g ( Y )) m ( A ) (15) 1 P F p F 1 A g ( Yp) Pl ( p p) Pl ( h ( Y )) m ( A ) (16) 1 P F p F 1 A ( Yp) Where Y { p : p p, p } p P From (15) ad (16), the lower ad upper bouds o system alure probablty ca be obtaed, where Bel ( p p) s the lower boud ad Pl ( p p) s the upper boud P P 4 Numercal Example For seres-parallel system Fg 4, basc bele assgmet (BBA) o each ut s show table 1 For smplcty, suppose ut 1 s same as ut 2, ad ut 3 same as ut 4 Table IBBA o seres-parallel system F 1, F 2 m(x 1) F 3, F 4 m(x 2) [0003,0004] 05 [0003,0005] 04 [0002,0005] 03 [0004,0006] 03 [0002,0007] 01 [0002,0006] 02 [0001,0008] 01 [0002,0007] 01 Form (14), system alure probablty ucto s mootoe decreasg So wth (15) ad (16), lower ad upper bouds o alure probablty P s obtaed by vertex method [16], as show Fg From Fg 2, system alure probablty P s the terval [799 10, ], ad probablty o 5 P s 003, probablty o P 5 4 [150 10, ] s 098, so P belog to terval wth codece o 95% I terval aalyss s used to calculate ths ssue [8-10], the 6 4 result s that P s the terval [799 10, ], whch s smlar wth D-S evdece method But there s o addtoal ormato ca be obtaed 1 08 Bele Plausblty 06 P(<173e-4)= P(<150e-5)= p 10-4 Fg 5 Probablty box o alure probablty o Seres-parallel system

6 84 Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty 5 Coclusos I the stuato that ut alure probablty s mprecse whe calculato the alure probablty o system, classcal probablty method s ot applcable, ad the aalyss result o terval method s coarse To calculate the relablty o seres-parallel systems above stuato, D-S evdece theory was used to represet the ut alure probablty Mult-sources ormato was used, ad bele ad plausblty ucto were used to calculate the relablty o seres-parallel systems by evdetal reasog By ths mea, lower ad upper bouds o probablty dstrbuto o system alure probablty were obtaed The smulato result shows that the proposed method s preerable to deal wth the mprecse probablty relablty calculato, ad ca get addtoal ormato whe compare wth terval aalyss method I uture work, back-up redudacy, k-out-o- ad cosecutve k-out-o- system, whch basc uts are ucerta, should be studed Also the alure probablty uctos o these systems are deret rom seres ad parallel systems, but the method to deal wth these ucertates are same by usg D-S evdece theory Reereces [1] Be-Ham Y, Covex models o ucertaty radal pulse bucklg o shells, Joural o Appled Mechacs, vol 60, o 3, pp , March 1993 [2] Elshako I, Essay o ucertates elastc ad vscoelastc structures: From AM Freudethal's crtcsms to modem covex modelg, Computers ad Structures, vol 56, 6, pp , Jue 1995 [3] Be-Ham Y, A o-probablstc cocept o relablty, Structural Saety, vol 14, 4, pp , Aprl 1994 [4] Yakov Be-Ham, Ucertaty, probablty ad ormato-gaps, Relablty Egeerg ad System Saety, vol 85, 2, pp , February 2004 [5] S Gareth Perce, Yakov Be-Ham, et al, Evaluato o Neural Network Robust Relablty Usg Iormato-Gap Theory, IEEE Trasactos o Neural Networks, vol 17, 6, pp , Jue 2006 [6] Scott J Duca, Bert Bras, Chrstaa JJ Pareds, A approach to robust decso makg uder severe ucertaty le cycle desg, Iteratoal Joural o Sustaable Desg, vol 1, 1, pp 45-59, Jauary 2008 [7] Qu Zhpg, Che Suhua, Elshako I, No-probablstc egevalue problem or structures wth ucerta parameters va terval aalyss, Chaos, Soltos ad Fractals, vol 7, 3, pp , March 1996 [8] Wedog Wu, SS Rao, Ucertaty aalyss ad allocato o jot toleraces robot mapulators based o terval aalyss, Relablty Egeerg ad System Saety, vol 92, 1, pp 54-64, Jauary 2007 [9] C Jag X Ha, F J Gua, Y H L, A ucerta structural optmzato method based o olear terval umber programmg ad terval aalyss method, Egeerg Structures, vol 29, 11, November 2007 [10] Fahed Abdallah, Amadou Gg, Phlppe Boat, Box partcle lterg or olear state estmato usg terval aalyss, Automatca, vol 44, 3, March 2008 [11] D Harmace, G J Klr, Measurg total ucertaty Dempster-Shaer theory: A ovel approach, Iteratoal Joural o Geeral Systems, vol 22, 4, pp , Aprl 1997 [12] R R Yager, Arthmetc ad other operatos o Dempster-Shaer structures, Iteratoal Joural o Ma-mache Studes, vol 25, pp , Jue 1986 [13] Dempster A P, Upper ad lower probabltes duced by a mult valued mappg, Aals Math Statst, vol 38, o 2, pp , 1967 [14] Shaer G A, Mathematcal Theory o Evdece Prceto Uversty Press, 1976 [15] C Josly, JC Helto, Bouds o bele ad plausblty o uctoalty propagated radom sets, 2002 Aual Meetgs o the North Amerca Fuzzy Iormato Processg Socety, pp27-29, Jue 2002, New

7 Calculato o Falure Probablty o Seres ad Parallel Systems or Imprecse Probablty 85 Orleas, LA, IEEE, Pscataway, NJ, pp: , 2002 [16] Harsh Agarwal, Joh E Reaud, Eva L, Presto, et al Ucertaty quatcato usg evdece theory multdscplary desg optmzato, Relablty Egeerg ad System Saety, vol 85, 1, pp , Jauary 2004