Remenyte-Presott, Rs nd Andrews, John (27) Prime implints for modulrised non-oherent fult trees using inry deision digrms. Interntionl Journl of Reliility nd Sfety, (4). pp. 446-464. ISSN 479-393 Aess from the University of Nottinghm repository: http://eprints.nottinghm..uk/336//prime_implints_for_modulrised_nonoherent_fult_trees.pdf Copyright nd reuse: The Nottinghm eprints servie mkes this work y reserhers of the University of Nottinghm ville open ess under the following onditions. This rtile is mde ville under the University of Nottinghm End User liene nd my e reused ording to the onditions of the liene. For more detils see: http://eprints.nottinghm..uk/end_user_greement.pdf A note on versions: The version presented here my differ from the pulished version or from the version of reord. If you wish to ite this item you re dvised to onsult the pulisher s version. Plese see the repository url ove for detils on essing the pulished version nd note tht ess my require susription. For more informtion, plese ontt eprints@nottinghm..uk
Prime Implints for Modulrised Non-oherent Fult Trees Using Binry Deision Digrms Rs Remenyte-Presott, John D. Andrews Aeronutil nd Automotive Engineering, Loughorough University, Loughorough, UK Keywords Fult Tree Anlysis, Binry Deision Digrms, Non-oherent systems, prime implints Astrt This pper presents n extended strtegy for the nlysis of omplex fult trees. The method utilises simplifition rules, whih re pplied to the fult tree to redue it to series of smller sutrees, whose solution is equivlent to the originl fult tree. The smller sutree units re less sensitive to the si event ordering during BDD onversion. BDDs re onstruted for every sutree. Qulittive nlysis is performed on the set of BDDs to otin the prime implint sets for the originl top event. It is shown how to extrt the prime implint sets from omplex nd modulr events in order to otin the prime implint sets of the originl fult tree in terms of si events.. Introdution Fult tree nlysis ws first oneived in the 96's nd provides good representtion of the system from n engineering viewpoint. This form of the filure logi funtion does not however lend itself to esy nd urte mthemtil mnipultion. A more onvenient form for the logi funtion from the mthemtil viewpoint is tht of Binry Deision Digrm [,2]. It overomes some disdvntges of onventionl FTA tehniques enling effiient nd ext qulittive nd quntittive nlyses of oth oherent nd non-oherent fult trees. The diffiulty of the BDD method is with the onversion proess of the fult tree to the BDD. An spet of this is tht the si events in the fult tree hve to e pled in n ordering required to onstrut the BDD. A good ordering gives onise BDD form. A d ordering my led to n explosion in the size of the BDD used to represent the fult tree. To dte n optimum strtegy for produing BDDs for ll fult trees hs not een identified. Non-ohereny introdues diffiulties in the qulittive ssessment of the fult tree ompred with the oherent version. The filure modes of the system, omintions of working or filed omponents whih result in system filure, tend to e greter in numer nd lrger in order thn the oherent version. This n render the qulittive evlution intrtle. The BDD method is more effiient for nlysing system without the need for the pproximtions used in the trditionl pproh of kineti tree theory [3]. In this pper two simplifition strtegies tht hve een shown to e effetive in reduing the omplexity of the prolem re pplied: redution [4] nd modulristion
[5]. The redution tehnique simplifies the fult tree to its miniml logi form, whilst modulristion reks down the fult tree to independent sutrees tht n e nlysed seprtely. Then BDDs re otined for eh module in seprte omputtions, ulminting in set of BDDs, whih together represent the originl system filure digrm. A qulittive nlysis of non-oherent fult trees using BDDs is the sujet of this pper. Met-produts BDDs [6,7] used to determine the prime implint sets re then developed. Then prime implint sets for every module n e lulted nd extrted for the whole system. Eh of these stges is desried in detil in the following setions nd demonstrted throughout y the use of n exmple. 2. Non-oherent fult tree struture In oherent fult tree eh omponent in the system is relevnt, nd the struture funtion is monotonilly inresing [8]. A fult tree tht ontins only AND gtes nd OR gtes is lwys oherent. Whenever NOT logi gte is introdued into fult tree, it is likely to eome non-oherent. A fult tree is non-oherent when oth omponent filure nd working sttes (positive nd negtive events) ontriute use of the top event. For exmple, system filure might our due to the reovery of filed omponent. Alterntively, during system filure, the filure of n dditionl omponent my ring the system to good stte. Consider n exmple of the trffi light system [8] shown in Figure. RED G R E E N A B C Figure. Trffi light system Crs A nd B re pprohing the lights on RED nd should stop. Cr C is pprohing the juntion with lights on GREEN nd should proeed. Given this senrio, the following filure events n our: A - Cr A fils to stop B - Cr B fils to stop C - Cr C fils to ontinue An ident t this rossrods n our in two wys: Cr A ts properly nd stops ( A ) nd r B fils to stop (B) nd drives into the k of A. Cr A fils to stop (A) nd if r C ontinues to move into the rossing ( C ), A hits C. The fult tree to represent the uses of n ident sitution t this rossrods is illustrted in Figure 2.
Aident t rossrods TOP Cr A ollides with r C G Cr B ollides with r A G2 Figure 2. Fult tree for trffi light system Working in top-down wy the following logi expression is otined, whih is in disjuntive norml form (miniml sum of produts) Top A C A B. () where + is OR, is AND. A qulittive nlysis of the non-oherent fult tree will produe the system filure modes known s prime implint sets. Prime implints re defined s omintions of omponent onditions (working or filed) whih re neessry nd suffiient to use system filure. Tking the produed terms from eqution gives prime implint sets A C nd A B. However, unlike the redution of the logi expression for oherent fult tree the logi eqution for the top event in the non-oherent se does not produe omplete list of ll prime implints. In this exmple, if r B fils to stop nd r C ontinues ross the lights there will e ollision whtever A does. Therefore, the full logi expression for the Top event is: Top A C A B B C, (2) whih n e otined y pplying the onsensus lw: A C A A X A Y A X A Y X Y. (3) This exmple is inluded just s mens to demonstrte issues for non-oherent fult trees. The inrese in length of the logi eqution for non-oherent fult trees n use diffiulties for lrge fult trees nd is t times neessry [8]. The inlusion of NOT logi gtes is used in Event Tree Anlysis where there re omponent filures tht our in more thn one fult tree whih represent the uses of the rnhing [9]. Also, NOT logi is importnt in Phsed Mission Anlysis []. It is possile to redue the prime implint sets to their oherent pproximtions prior to the fult tree quntifition. This is hieved y removing ll suess sttes nd reminimising the expression (on the sis of likelihood is lose to ). This will give miniml ut sets {A} nd {B}. A A B
3. Simplifition of the fult tree struture Deling with omplex industril systems n result in very lrge fult trees, whose nlysis is time onsuming. Two pre-proessing tehniques n e pplied to the fult tree in order to otin the smllest possile sutrees nd redue the size of the prolem. The first prt of the simplifition proess is redution tehnique [4] whih resizes the fult tree to its simplest form. The seond prt identifies independent modules (sutrees) within the fult tree tht n e delt with seprtely [5]. The liner-time lgorithm is pplied to the seond prt nd set of independent fult trees in their simplest possile struture is otined. It is equivlent to the originl system filure uses nd is esier to mnipulte during the nlysis proess. The nlysis strtegy of oherent fult trees inorporting the simplifition proess ws presented in []. This pper provides development of this tehnique in the non-oherent se. 3.. Redution This tehnique redues the fult tree to its simplest form while retining its logil struture. It is pplied in four stges: ontrtion, ftoristion, extrtion, sorption. First of ll, the fult tree is mnipulted so tht the NOT logi is pushed down the fult tree until it is pplied to si events using De Morgn s lws, i.e. A B A B (4) A B A B (5) Then for the ontrtion, susequent gtes of the sme type re ontrted to form single gte so tht the fult tree eomes n lternting sequene of AND nd OR gtes. During the ftoristion, pirs of events tht lwys our together s inputs to the sme gte type re identified nd omined forming single omplex event. If events pper in their working nd filed sttes in the fult tree, only those si events tht pper together in their negted stte under the opposite gte type n e omined. (Note: in this sense the AND type gte is opposite to the OR type gte). By De Morgn s equtions 4 nd 5, if nd/or pper in the fult tree, then forms omplex event, or if nd/or pper in the fult tree, then forms omplex event. The omplex events identified re then sustituted into the fult tree struture. In the extrtion stge, the two strutures shown in Figure 3 re identified nd repled in order to redue the repeted ourrene of events to single ourrene nd filitte further redution.
restruture restruture (i) (ii) Figure 3. Redution, the extrtion proedure If nother omponent is repeted in the struture nd it is repeted in its negted stte, the strutures shown in Figure 4 n e simplified even more, i.e. the whole struture is repled y the omponent tht ppers only in one stte, filed or working. restruture restruture (i) (ii) Figure 4. Redution, the extrtion proedure in non-oherent se During sorption, strutures were identified tht ould e further simplified through the pplition of the sorption nd idempotent lws to the fult tree logi. The simplifition proess n e pplied when n event is repeted in the rnh of the fult tree struture. The gte with the first ourrene of the repeted event is lled primry gte nd gte with the seond ourrene of the repeted event is lled seondry gte. There n e two types of the repetition in non-oherent system, i.e. omponent n e repeted in the sme stte or it n e repeted in its negted stte. Considering the first se, if the primry nd the seondry gtes with n event in ommon re of different type (Figure 5(i)), the struture is simplified y removing the whole seondry gte nd its desendnts. If the primry nd the seondry gtes re of the sme type (Figure 5(ii)), the struture is simplified y deleting the ourrene of the event eneth the seondry gte.
d (i) (ii) Figure 5. Redution, the sorption proedure If omponent is repeted in its negted stte, the sorption rule n gin e pplied. In this se the sorption nnot e pplied if the primry gte is n OR gte. Therefore, if the primry gte is n AND gte nd the seondry gte is n OR gte (Figure 6(i)), then the struture is simplified y deleting the ourrene of the event eneth the seondry gte. If oth the primry gte nd the seondry gte re AND gtes (Figure 6(ii)), the whole seondry gte n e deleted. The order of pperne of positive nd negtive events in primry nd seondry gtes is irrelevnt. d d (i) (ii) Figure 6. Redution, the sorption proedure in non-oherent se The ove four steps re repeted until no further hnges tke ple in the fult tree. An exmple of the pplition of these rules to fult tree is demonstrted onsidering the fult tree shown in Figure 7.
Top Top G 2 G d G2 G3 2 22 G2 e h G4 G5 h e 22 G4 i f h f e f G5 Figure 7. Exmple fult tree Top 22 i Figure 8. Redued fult tree 2 G 2 22 G2 22 f Figure 9. Redued fult tree 2
Contrtion ws performed for gtes G nd G3, whih re of the sme type, forming single gte. Ftoristion ws performed three times: for the pir of si events AND for the pir of si event OR d nd for the pir of si events e AND h, reting omplex events 2, 2 nd 22 respetively. Bsi events e OR h form omplex event 22. The orresponding omplex event dt re shown in Tle. The extrtion proedure ws pplied to gtes G4 nd G5, extrting the repeted event f. The simplified fult tree is shown in Figure 8. Finlly, the sorption ws pplied to gte G2 removing gte G5. No further simplifition is possile. The finl redued tree is shown in Figure 9. Tle. The omplex event dt Redution hs simplified the exmple fult tree. In the originl tree there were six gtes, in the redued tree there re 3 gtes. There were thirteen events in the originl fult tree, eight of them different; in the redued tree there re five events, nd four of them re different. For lrge systems the degree of simplifition is fr more signifint. Hving redued the fult tree to more onise form, the seond pre-proessing tehnique of modulristion is onsidered. 3.2. Modulristion Complex event Gte vlue Event Event 2 2 AND 2 OR d 22 AND e h The modulristion tehnique identifies independent sutrees within the fult tree. Every module ontins no si events tht pper elsewhere in the fult tree nd it n e nlysed seprtely. The results from eh module re sustituted into the higher level fult trees where modules our nd further stges of the nlysis re performed. For the purposes of the proess si events in their positive or negtive form re onsidered s the sme entity s it is only the vent lel tht is importnt in identifying independent modules. Using the liner-time lgorithm [5] the independent modules n e identified fter just two depth-first trversls of the fult tree. During the first time of the trversl the step numer t the first, seond nd finl visits to every node (gte nd event) is reorded. Then the mximum (Mx) of the lst visits nd the minimum (Min) of the first visits of the desendnts (ny gtes or events ppering elow tht gte) of eh gte re lulted. Step numers for every node in the exmple fult tree, Mx nd Min of the gtes nd events for the redued tree in Figure 9 re presented in Tles 2, 3 nd 4 respetively.
Step numer 2 3 4 5 6 7 8 9 Node Top 2 G 2 22 G2 22 f G2 G Top Tle 2. Step numers for every node in the fult tree Gte Top G G2 st visit 2 nd 3 6 visit 9 Finl visit 9 Min 2 4 5 Mx 9 8 Tle 3. Dt for gtes in the fult tree Event 2 2 st visit 2 nd visit Finl visit 22, 22 2 4 5 8 2 4 7 8 2 4 7 8 Tle 4. Dt for events in the fult tree f The rules for identifying gte s heding module re: The first visit to eh desendnt of gte is fter the first visit to the gte nd The lst visit to eh desendnt of gte is efore the seond visit to the gte. The following gtes n e identified s heding modules: Top, G. Gte G2 n not e module euse one its desendnt (event 22) is visited efore visiting prent gte. Gte G is repled y modulr event M. Two seprte fult trees, shown in Figure, now reple the fult tree in Figure 9. Top M : G 2 M 2 22 G2 22 22 f Figure. The two modules otined for the fult tree shown in Figure 9
Hving redued the fult tree to its miniml form nd identified ll the independent modules the next stge is to otin the BDDs. 4. Conversion to inry deision digrms Every independent fult tree is onverted to BDD. The vrile ordering needs to e estlished efore the onversion proess. In this pper the vrile ordering sheme for every module is set to e left-right top-down. For exmples s smll s these the vrile ordering is lrgely irrelevnt. Following the hosen sheme gives the orderings of si events: Top: 2 < M, M: 2 < 22 < f, The BDD onstrution lgorithm is desried in referene [2]. The resulting SFBDD enodes the struture funtion of non-oherent fult tree. The kground of it is to ssign n ordered triple to eh node in the SFBDD: ite(x,f,f ), (6) here x is the Boolen vrile representing si event s the node nd f nd f re the logi funtions on its rnh nd rnh respetively. ite is desried s ifthen-else opertion, i.e. If x ours then onsider f, else onsider f. endif SFBDD onstrution then moves through the fult tree in ottom-up mnner. Bsi events re ssigned ite strutures. For exmple, si event is expressed s: = ite(,,). (7) Alterntively, si event is ssigned n ite struture: = ite(,,). (8) When deling with gtes their input events, J nd H, re expressed in the ite form, the following rules re pplied: If J = ite(x,f, f ) nd H = ite(y,g, g ), ifre( x, K, K, K K) then J H ifre( x, L, L, L L ) if x y in theordering, (9) if x y in theordering. Applying these rules to the fult tree in Figure results in the set of SFBDDs presented in Figure.
Top: 2 F M : 2 F3 F2 M F4 22 F5 f Figure. The otined SFBDDs for the modules shown in Figure The qulittive nd quntittive nlyses n e rried out sine the omplete set of SFBDDs hve een omputed. Now the lultion of prime implint sets will e presented using the SFBDDs otined from simplified fult trees. 5. Clultion of prime implint sets Knowledge of prime implint sets n e vlule in gining n understnding of the system. It n lso help to develop repir shedule for filed omponents if system nnot e tken off line for repir. The SFBDD whih enodes the struture funtion nnot e used diretly to produe the omplete list of prime implint sets of nonoherent fult tree. For exmple, onsider generl omponent x in non-oherent system. Component x n e in filed or working stte, or n e exluded from the filure mode. In the first two situtions x is sid to e relevnt, in the third se it is irrelevnt to the system stte. Component x n e either filure relevnt (the prime implint set ontins x) or repir relevnt (the prime implint set ontins x ). A generl node in the SFBDD, whih represents omponent x, hs two rnhes. The rnh orresponds to the filure of x; therefore, x is either filure relevnt or irrelevnt. Similrly, the rnh orresponds to the funtioning of x nd so x is either repir relevnt or irrelevnt. Hene it is impossile to distinguish etween the two ses for eh rnh nd the prime implint sets nnot e identified from the SFBDD. There is numer of tehniques developed for lulting prime implint sets. The first pproh ws presented y Courdet nd Mdre [6] nd further developed y Dutuit nd Ruzy [7] where met-produts BDD is formed. This tehnique is used in the pper. Some other lterntive tehniques for lulting prime implint sets were presented y Sso in [2], y Ruzy in [3] nd y Contini in [4]. They use lterntive nottions for the representtion of prime implints. In met-produts BDD method developed n lterntive nottion is developed tht ssoites two vriles with every omponent x. The first vrile, P x, denotes relevny nd the seond vrile, S x, denotes the type of relevny, i.e. filure or repir relevnt. A met-produt, MP(π), is the intersetion of ll the system omponents ording to their relevny to the system stte nd π represents the prime implint set enoded in met-produt MP(π).
Px S x if x MP( ) Px S x if x () Px if neither x nor x elongs to The proposed lgorithm is used for lulting the met-produts BDD of fult tree from the SFBDD. The met-produts BDD is lwys miniml, therefore it enodes the prime implint sets extly. In order to present the lgorithm, onsider node F in SFBDD, where F = ite(x, F, F). The met-produts BDD, tht desries prime implint sets using eqution 2, is expressed s: PI(F) = ite(p x, ite(s x, P, P), P2), () where P2 = PI(F F), (2) P = PI(F) P 2, (3) P = PI(F) P 2. (4) x is the first element in the vrile ordering, PI(F) represents the struture of metproduts BDD. P2 enodes the prime implints for whih x is irrelevnt, P enodes the prime implints for whih x is filure relevnt nd P enodes the prime implints for whih x is repir relevnt. If not ll the si events in the vrile ordering pper on the prtiulr pth, then PI(F) = ite( P,, PI(F)). (5) x j here x j is efore x i nd x j does not pper on the urrent pth from the root node to node F. If F is terminl node, then PI() = ite( P x i,, ite( P PI() =, (6) x i,,, ite( P,, ))). (7) where x i,, x n re si events from the vrile ordering tht hve not yet een onsidered in the proess. *** Consider the BDD for Top module in Figure. Applying equtions -4 to node F gives: PI(F) = ite(p 2, ite(s 2, PI(F2), PI() ), PI(F2 )) = ite(p 2, ite(s 2, PI(F2), ), ). In the sme wy pplying equtions -4 to node F2 gives: PI(F2) = ite(p M, ite(s M,, ), )). The met-produts BDD is otined for Top module. x n
Then the BDD for module M is investigted. Applying eqution 3 to node F3 gives: PI(F3) = ite(p 2, ite(s 2, PI() PI F4, PI(F4) PI F4, PI(F4)). First of ll, PI() is lulted. Sine there re two more vriles in the ordering sheme tht were not investigted yet, i.e. 22 < f, ording to eqution 7 we get: PI() = ite(p 22,, ite(p f,, )). Then PI(F4) is lulted pplying equtions -4: PI(F4) = ite(p 22, ite(s 22, PI(F5) PI F5, PI() PI F5, PI(F5)). PI() is lulted in the sme. Sine there is only one vrile left in the ordering sheme, i.e. vrile f, ording to eqution 7 it gives: PI() = ite(p f,, ). PI(F5) is otined pplying equtions -4 to node F5. i.e. PI(F5) = ite(p f, ite(s f,, ), ). During the negtion of PI(F5) terminl verties nd verties re swpped in the met-produts BDD. Therefore, PI F5 = ite(p f, ite(s f,, ), ). Sine there re no repeted prts etween PI() nd PI F5, PI() PI F5 = PI(). Then the lultion of PI(F4) is ompleted: PI(F4) = ite(p 22, ite(s 22,, ite(p f,, )), ite(p f, ite(s f,, ), )). PI F4 is otined in the sme wy, swpping terminl verties nd. Beuse there re no repeted prts etween PI() nd PI F4, PI() PI F4 = PI(). This ssures tht the met-produts BDD produed is in its miniml form. The lultion of the met-produts BDD for module M is ompleted. The set of resulting met-produts BDDs for two fult trees in Figure is shown in Figure 2.
P 2 P 2 S 2 S 2 P M P 22 P 22 S M P f P f S f (i) (ii) S 22 P f Figure 2. Met-produts BDDs for lulting prime implint sets Now it is possile to otin the met-produts nd identify the prime implint sets. First module Top P S P S 2 2 M M { 2, M} Seond module M P P P 2 S2 P22 Pf 2 P22 S22 Pf P P S { 2 22 f f f {2} {22} }
For exmple, in the seond met-produt P 2 signifies tht omponent 2 is relevnt nd S 2 signifies tht omponent 2 is filure relevnt. P 22 nd P f men tht omponents 22 nd f re irrelevnt. Hene the prime implint otined is {2}. The prime implint sets hve een produed for every independent module, tht ontin modulr nd omplex events. It is essentil to e le to nlyse the system in terms of its originl omponents, therefore, the next stge of the qulittive nlysis hs to onsider the extrtion of the omintions of omponent filures from every omplex nd modulr event. A key point of the expnsion lgorithm, whih is the sme s the MOCUS method [5] for lulting miniml ut sets from fult trees, is tht n AND gte inreses the numer of si events in eh prime implint set nd n OR gte inreses the numer of prime implint sets in the system. A two dimensionl rry is reted. Eh line in the rry represents prime implint set. At the strt the top event gte is loted in the first row nd the first olumn of the two-dimensionl rry. Then repetedly the rry is snned repling:. Eh omplex event whih is n OR gte y vertil expnsion inluding the input events to the gte (dupliting ll other events in this row) 2. Eh omplex event whih is n AND gte y horizontl expnsion inluding the input events to the gte 3. Eh modulr event (originl or negted) y vertil nd/or horizontl expnsion inluding the list of prime implint sets otined from the met-produts BDD, whih represents the modulr event (originl or negted until only si events pper in the rry. If prime implint sets need to e produed for the negted modulr event, the originl BDD for this event need to e negted, repling its terminl verties to verties nd the other wy round. Then the met-produts BDD for the negted modulr event n e omputed using the lgorithm presented in referenes [6,7]. Clultion of prime implint sets using this lgorithm ws performed nd shown in Figure 3.
Top 2 M M (i) (ii) (iii) 2 22 d f e (iv) h f Figure 3. Extrting prime implint sets from modulr nd omplex events First module Top hs produed one prime implint set {2, M }, whih reple the top event in the rry, s shown in Figure 3(ii). Sine omplex event 2 = AND, its inputs nd reple the gte in horizontl expnsion. The otined rry is shown in Figure 3(iii). Also, module M n e repled. M produes three prime implint set {2}, { 22 }, {f}. This set results in vertil expnsion in the rry. The resulting rry is shown in Figure 3(iv). Then the expnsion of the inputs for omplex event 2 nd event 22 give two more horizontl expnsions, shown in Figure 3(v). The prime implint sets in the rry ontin only si events, therefore the lultion is finished. The prime implint sets of the exmple fult tree, presented in Figure 7 re: {,,}, {,,d}, {,, h }, {,, e }, {,,f}. 6. Comprison of qulittive nlyses using originl nd simplified fult trees An nlysis hs een onduted on the fult tree to BDD onversion proess. In this nlysis some exmple fult trees were onverted to BDDs nd then qulittive nlysis performed. Sixteen exmple fult trees were nlysed y pplying the BDD method to oth the originl nd the simplified fult trees. Tle 5 provides summry of the results for eh fult tree. (v)
Exmple Numer of gtes Numer of si events Numer of omplex events Numer of modules Numer of nodes in BDD with simplifitions Numer of nodes in BDD without simplifitions Numer of prime implint sets Time tken with simplifitions Time tken without simplifitions 3 46 23 5 88 855 43.72 66.86 2 7 58 39 533 7229 4238.64 434.782 3 7 63 35 3738 584 5327.828 9.729 4 5 69 44 3824 59498 357 7.562 573.39 5 2 6 28 4769 2.687.938 6 27 2 2 4452 7459 38.72.69 7 39 27 433 6666 5 8.25 9.734 8 27 6 3 5765-373 9.29-9 32 66 9 6646-67 8.42-22 55 27 583 683695 339.859 43.425 26 48 8 94827 33624 47 33.295 63.488 2 32 38 4 3656 367 5.844 3.297 3 27 57 28 323 53393 23 4.42 796.779 4 2 29 3 3464 9767 46.63 6.653 5 25 4 7 27 7725 74.547 8.92 6 2 35 9 34633 34464 64 6.64 99.27 Tle 5. Clultion results for exmple fult trees The seond nd the third olumns give some inditions of the omplexity of the exmple fult trees, presenting the numer of gtes nd si events. The results of the two simplifition tehniques re shown in the fourth nd fifth olumns, whih represent the numer of omplex nd modulr events respetively. The redution tehnique hs redued the size of the prolem remrkly, espeilly for exmples 2, 3 nd 4. The modulristion tehnique extrted five independent modules for exmple. No more modules were extrted for other exmples sine the omplex ftors hd lredy redued the tree struture to its most effiient form. The sixth nd seventh olumns show the numer of nodes in BDDs, whih were otined using the simplified nd the originl fult tree dt respetively. A sum of numer of nodes in SFBDDs nd met-produts BDDs is exmined sine oth types of BDDs re used in the qulittive nlysis of non-oherent fult trees. The simplifition proedure deresed the size of the BDDs signifintly. The numer of nodes deresed y pproximtely one hlf t lest when the simplifition rules on the fult trees were pplied. Extrtion of modules nd omplex events hd ruil effet on the iggest trees (exmples 8 nd 9) euse it enled the onversion proess of fult trees to BDDs, wheres due to the size of the BDDs, the proess filed if the originl fult tree strutures were used. BDDs ould not e formed in the memory resoures ville. The eighth olumn represents the numer of prime implint sets in the solution. This gin indites the omplexity of the prolem. The lst two olumns respetively give the time tken to perform the nlysis if simplified nd originl fult trees were used.
The time deresed when simplifition rules were pplied euse smller BDDs were otined. Sine the onversion proess for exmple 8 nd 9 filed, the entries for the time re not reported euse nlysis ws unle to e performed. 7. Conlusions This pper presents proedure y whih lrge fult trees n e simplified prior to onversion to their Binry Deision Digrm form for nlysis. Non-oherent fult trees re exmined. Simplifition is performed in two phses, the first redues the fult tree to its more onise form nd retins the underlying prolem struture. The seond phse identifies independent modules whih n e nlysed seprtely. In doing this the prolem n e solved effiiently. Hving performed the simplifition the prolem is solved in terms of the new modulr struture nd omplex events. A mens of lulting prime implint sets in terms of the originl si events is presented. The effiieny of the simplifition proess is nlysed using some exmple fult trees nd it is ompred with the nlysis of fult trees tht were not simplified. A omprison of two methods illustrtes fvourle degree of effiieny in the simplifition of fult trees prior to onversion to their BDD form for nlysis. 8. Referenes. Brynt, R.E. Grph-Bsed Algorithms for Boolen Funtion Mnipultion', IEEE Trns. Computers, C-35, No.8, pp677-69, (986) 2. Ruzy, A. New Algorithms for Fult Tree Anlysis, Reli Eng Syst Sfety, 4, pp23-2 (993). 3. Vesely, W.E. A Time Dependent Methodology for Fult Tree Evlution, Nuler Design nd Engineering, 3, pp337-36, (97). 4. Pltz, O. nd Olsen, J.V. FAUNET: A progrm Pkge for Evlution of Fult Trees nd Networks, Reserh Estlishment Risk Report, No. 348, DK-4 Roskilde, Denmrk, Sept. (976) 5. Dutuit, Y. nd Ruzy, A. A Liner-Time Algorithm to Find Modules of Fult Trees, IEEE Trns. Reliility, 45, No.3, pp422-425, (996) 6. Courdet, O. nd Mdre, J.-C. A New Method to Compute Prime nd Essentil Prime Implints of Boolen Funtions, Advned Reserh in VLSI nd Prllel Systems, pp3-28 (992) 7. Dutuit, Y. nd Ruzy, A. Ext nd Trunted Computtions of Prime Implints of Coherent nd Non-Coherent Fult Trees with Arli, Reliility Engineering nd System Sfety, 58, pp225-235 (997) 8. Andrews, J.D. To Not or Not to Not!!, Proeedings of the 8 th Interntionl System Sfety Conferene, pp267-274 (2) 9. Andrews, J.D. nd Dunnett, S.J. Improved Aury in Event Tree Anlysis, Foresight nd Preution. Cottm, Hrvey, Ppe nd Tte (eds). Proeedings of ESREL 2, SARS nd SRA-EUROPE nnul onferene, pp525-532, (2). L Bnd, R.A. nd Andrews, J.D. Phsed Mission Modelling Using Fult Tree Anlysis, Proeedings of the IMehE Prt E Journl of Proess Mehnil Engineering, 28, pp83-9, (24). Rey, K.A. nd Andrews, J.D. A Fult Tree Anlysis Strtegy Using Binry Deision Digrms, Reliility Engineering nd System Sfety, 78, pp45-56, (22)
2. Sso, T. Ternry Deision Digrms nd their Applitions, Representtions of Disrete Funtions, Chpter 2, pp269-292 (996) 3. Ruzy, A. Mthemtil Foundtion of Miniml Cutsets, IEEE Trnstions on Reliility, volume 5, 4, pp389-396 (2) 4. Contini, S. Binry Deision Digrms with Lelled Vriles for Non-Coherent Fult Tree Anlysis, Europen Commission, Joint Reserh Centre (25) 5. Andrews, J.D. nd Moss, T.R. Reliility nd Risk Assessment, Professionl Engineering Pulishers, (22)