Integration of Dimensional Quality and Locator Reliability in Design and Evaluation of Multi-station Body-In-White Assembly Processes

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1 Inegraon of Dmensonal Qualy and Locaor Relably n Desgn and Evaluaon of Mul-saon Body-In-Whe Assembly Processes Yong Chen Deparmen of Mechancal and Indusral Engneerng, he Unversy of Iowa Iowa Cy, IA Jonghua Jn 1 Deparmen of Sysem & Indusral Engneerng, he Unversy of Arzona ucson, Arzona Janun Sh Deparmen of Indusral & Operaons Engneerng, he Unversy of Mchgan Ann Arbor, MI Absrac: Research effors have been made n he developmen of a Qualy and Relably Chan (QR-Chan model o negrae manufacurng sysem componen relably and produc ualy n mul-saon manufacurng processes. Based on a prevously developed sae space model, whch capures he varaon propagaon hroughou all saons, a general QR-Chan model s newly developed for Body-In-Whe (BIW assembly processes. he effecveness of he QR-Chan modelng sraegy s demonsraed by horoughly sudyng he relaonshp beween locaor performance and produc ualy n assembly processes. Based on he analycal resul of sysem relably obaned from he QR-Chan model, opmal locaor wear rae assgnmen s furher nvesgaed. A case sudy s conduced o demonsrae he effecveness and poenal usage of he QR-Chan model for he BIW assembly sysem desgn evaluaon and opmzaon. ey words: sae space model, BIW assembly process, relably modelng, QR-Chan, ualy and relably neracon 1 All correspondence should be addressed o Dr. J. Jn, Phone: ; Fax: ; E-mal: hn@se.arzona.edu 1

2 1. Inroducon An auomove body whou doors, hood, fenders and runk ld s called a Body n Whe (BIW. In a BIW assembly lne, dependng on he complexy of he produc, here are ypcally 8 o 13 assembly saons ha assemble 15 o 5 shee meal pars. Sysem relably of he BIW assembly process s one of he key facors affecng he producvy and produc ualy. In general, he sysem falure of a BIW assembly process ncludes boh he oolng componen caasrophc falure and he unsasfacory produc ualy. oolng componen caasrophc falures such as locang pn broken and loosen drecly lead o an mmedae sop of he auomaon process. Nonconformng produc ualy such as large produc varaon s an ndcaon ha he process has los s capably of producng producs wh he specfed ualy. In addon, real process daa have shown sgnfcan neracons beween he locang ool relably and he produc ualy propagaed hrough each saon of a BIW assembly process. For example, prevous research ndcaes ha 7% of he roo causes of dmensonal errors of a BIW are due o locang ool malfuncon (Ceglarek and Sh, 1995, whch ndcaes sgnfcan effecs of locang ool relably on he dmensonal produc ualy. On he oher hand, large dmensonal errors of he locang-holes on he ncomng produc may lead o locang ool caasrophc falures such as locang pn broken durng he par loadng process, par suck a pns, or par unable o be posoned correcly by he locaors. In hs paper, hese knds of locang ool falures are called locang ool falures nduced by ncomng produc ualy. herefore he caasrophc falure raes of locang ools are affeced by he dmensonal accuracy of he ncomng produc, whch s deermned by he propagaon of he dmensonal produc ualy from he prevous saons. Based on a prevous sudy n Yang e al. (, he

3 locang ool falure nduced by ncomng produc ualy corresponds o abou 44% of all locang ool caasrophc falures. herefore, he real process daa have shown srong neracons beween locang ool relably and produc ualy n a mul-saon BIW assembly process. In hs paper, relably of a BIW assembly process wh 3--1 fxures and rgd pars s suded. he rgd par assembly covers 68% of he oal pars n a ypcal auobody (Shu e al., he produc varaon s propagaed n a mul-saon assembly process (Jn and Sh, 1999; Dng e al.,. he fnal produc ualy s affeced by he accumulaon or sack up of all varaons generaed a prevous saons. he varaon propagaon n produc ualy wll lead o he propagaon of he neracon beween he locang ool relably and he produc ualy, whch s called as he QR-Chan effec. hs paper wll sudy he QR-Chan effec n a mul-saon BIW assembly process. A general QR-Chan modelng framework has been recenly proposed by Chen e al. (1 based on he process model, whch s n a general form of a lnear regresson model descrbng he relaonshp beween process varables and produc ualy n manufacurng processes. he process model plays a crcal role n analyzng he QR-Chan effec of a mulsaon manufacurng process (MMP, whch s assumed avalable n Chen e al. (1. However, s somemes no easy o oban a process model when Desgn of Expermens (DOE s no applcable, especally for a complex mulsage manufacurng process. hs paper wll propose a procedure o buld he process model for a mul-saon BIW assembly process based on he frs prncple of engneerng. Wh he process model obaned based on he process knowledge, he QR-Chan model by Chen e al. (1 can be successfully appled o he BIW assembly process. All parameers used n sysem relably analyss based he QR-Chan 3

4 model wll have he correspondng physcal meanngs. In addon, an applcaon of he QR- Chan model for opmal assgnmen of locaor wear rae s suded n hs paper for desgn mprovemen of BIW assembly processes. he process model of an assembly process should be obaned from he produc and process desgn nformaon and he physcal model of specfc processes wh consderaon of varaon propagaon. In recen years, fxure sysems and varaon propagaon n assembly processes have been suded and sgnfcan resuls have been acheved. he sascal descrpon of varaon paerns and he dagnosc ssues of a fxure sysem have been addressed (Hu and Wu, 199; Ceglarek e al., 1994; Ceglarek and Sh, 1996; Apley and Sh, For mulple saon assembly processes, Jn and Sh (1999 frs developed a sae space model o descrbe he produc varaon propagaon across dfferen saons. Based on hs sae-space model developmen, many research progresses have been recenly made n faul dagnoss (Dng e al., ; Dng e al., ; Zhou e al., 3, opmal sensor dsrbuon (Dng e al., 3b, and opmal process olerance desgn (Dng e al., 3a. However, none of he leraure above suded he assembly sysem relably and he neracons beween produc ualy and locang ool relably. In a recen research work (Jn and Chen, 1, he neracons beween produc ualy and locang ool relably are suded for a sngle assembly saon. he propagaon of he produc ualy and he QR-neracon n a mulsaon assembly process are no capured n Jn and Chen (1. In general, he QR neracon n a mul-saon assembly process s essenally much more complex han ha n a sngle saon process. Wh he ad of he sae space model developed n Jn and Sh (1999 o sudy he varaon propagaon, hs paper wll develop he correspondng process model for a mulsaon BIW assembly process and apply he QR-Chan model accordngly. 4

5 he paper s organzed as follows. he process varables and produc ualy characerscs n he conex of BIW assembly processes are specfed n Secon of hs paper. In Secon 3, he process model and oher elemens of he QR-Chan model are provded and he physcal meanngs of he model parameers are dscussed n he conex of he BIW assembly process. he sysem relably s evaluaed n Secon 4 based on he QR-Chan model for a BIW assembly process. he applcaon of he QR-Chan model s dscussed n Secon 5 o opmze he pn wear rae assgnmen. A case sudy s conduced n Secon 6 o demonsrae he proposed model and mehodology. he paper s concluded n Secon 7.. Revew of BIW Assembly Processes.1 Fxure Layou and Locang Prncple n BIW Assembly Processes In hs paper, a body coordnae sysem shown n Fgure 1 s used. he orgn of he body coordnae sysem s defned n he fron cener of a vehcle and below s underbody. he X-Y- Z axes are shown n he fgure. hs defnon of he body coordnae sysem has been wdely used n he auomove ndusry for produc and process desgn. Y O Z X. M1 P 1 M 3. B 1 B 3. M B z P y x (a Body coordnae sysem (b 3--1 fxurng prncple for a rgd par Fgure 1. Auomove body and s assembly fxure An assembly saon ypcally consss of wo or more assembly fxures. Each fxure holds a sngle par o be assembled wh oher pars. Locang pns and blocks are locang ools wdely used n fxures o deermne he par locaon and orenaon. In hs paper, a

6 fxure for rgd pars s assumed for each saon. As shown n Fgure 1(b, a ypcal 3--1 fxure conans several key componens: (1 a four-way pn/hole (P 1 o precsely locae a par n he X and Z drecons; ( a wo-way pn/slo (P o locae a par n he Z drecon; hese wo pns consran he par roaon and ranslaon n he X-Z plane ogeher; and (3 hree shaded locang blocks (B 1, B, B 3 o locae a par n he Y drecon. he combnaon of he locang ools (pns and blocks consrans all sx degrees-of-freedom of a rgd par. Snce he degree of wear-ou of locang blocks s very slgh compared o ha of locang pns, n hs paper we only consder he locang pns n BIW assembly processes.. Maor Elemens n he QR-Chan Framework of a BIW Assembly Process..1 Process Varables Assocaed wh Locang Pns A general modelng procedure focusng on he X-Z plane s presened n hs paper for rgd par assembly and he four-way and wo-way locang pns are consdered as sysem componens. Suppose here are n locang pns a he h saon, =1,,, L, where L s he number of saons n a BIW assembly process. he oal number of locang pns n all saons L n = 1 s n =. he changes of he pn dameer wll change he clearance beween he locang pn on he fxure and he locang-hole on he par, whch wll affec he produc ualy. hus he accumulaed decremen n he pn dameer due o pn wear-ou s consdered as he process varable. Le P, denoe he h locang pn a he h saon and X, ( denoe s accumulaed dameer decremen a me... Produc Qualy Characerscs n a BIW Assembly Process In assembly processes, produc ualy s generally defned by he dmensonal accuracy of he ey Produc Characersc (PC pons on he produc. he PC pons on he ougong produc of each saon nclude he locang-holes used for par locang n he nex saon and 6

7 he pons whose dmensonal accuracy s specfed n he produc desgn. he measuremens of hese PC pons are reaed as he produc characerscs n a BIW assembly process. Le Y = 1,..., m denoe he h produc characersc on he ougong produc of, 1,..., L, = saon a me, where m s he number of PC measuremens on he ougong produc of saon. 3. QR-Chan Model for BIW Assembly Processes here are several key relaonshps n he QR-Chan model of a BIW assembly process: he locang pn wear s he maor cause of PC devaons; produc ualy of each saon s defned based on he PC devaons; and he ougong produc ualy of a saon mpacs on he locang pn caasrophc falure of he nex saon. he followng dagram s used o summarze hese relaonshps: Pn DegradaonÆPC DevaonsÆProduc QualyÆPn Caasrophc Falure. he presenaon of hs secon wll follow he dagram shown above. 3.1 Locang Pn Degradaon Model he mechansm of he locang pn wear s dscussed n Jn and Chen (1. Accordng o Jn and Chen (1, he aggregaed wear of he pn dameer s ncreased wh he number of operaons, whch can be descrbed by a sochasc process model wh ndependenly lognormal dsrbued ncremens: X = X, ( 1 +, where ( s he random wear ncremen due o operaon, X ( s he nal clearance beween locang pn P, and s correspondng locang-hole, s he operaon ndex. I s assumed ha X ( ~ N( µ (, σ ( and,, 7

8 ~ LOGNOR( µ, (, σ, (, where µ ( and σ ( are he mean and sandard devaon of he lognormal random varable ( 1,1 ( µ 1, ( µ L, n L ( µ! and. Le [ ] σ 1,1 ( σ 1, ( % σ L., n L ( In hs paper, we choose one producon day as he me nerval o dscreze he me scale for a BIW assembly process. Le h denoe he number of operaons durng each producon day, and k denoe he end me of he k h producon day. Snce he me s measured by he number of operaons, k s he oal number of operaons unl he end of producon day k. So k =kh and =. he BIW assembly operaons are dscree n naure. he me of sldng wear when he par s posoned on he pn s much shorer han he cycle me of an operaon (more me s spen on weldng, clampng operaons and par handlng and movng. And he accumulaed wear of a locang pn s much smaller han he pn dameer and has lle mpac on fuure wear mechansm. So s reasonable o assume ha he wear amoun durng an operaon s ndependen of ha of prevous operaons. In addon, a BIW assembly process can produce 5-1,5 car bodes durng each day of producon. herefore he accumulaed wear of such a large number of operaons can reasonably be approxmaed as Normally dsrbued based on he cenral lm heorem. hus he followng euaon can be used o model he pn wear: where ~ N(, Q X ( = X( 1 +, k = 1,,... (1 k k k k ε, [ ] ε = h µ 1,1 ( µ 1, n (! µ L,1(! µ L, n ( 1!, L 8

9 ( σ (! σ (! σ (! σ ( Q = h dag, and X ~ N(,. In hs 1,1 1, n1 L,1 L, n L ( paper, ncreasng wear s assumed, ha s, for any 1 n, Pr{[X( ] <} and [ ( X( ] } Pr{ X 1 < can be gnored. k + k 3. Relaonshp beween Process Varables and PC Devaons In Chen e al. (1, he followng process model s used o descrbe he relaonshp beween process varables and produc ualy characerscs: Y =, +, +, z + X(, z, = 1,,..., L, = ( X1,1( X 1,! X L, n where [ ] n X 1,,..., m ( X R s he vecor of process varables n he L sysem; l z [ z1, z,..., zl ] R s he random vecor of nose varables, wh mean E(z and covarance marx Cov(z ndependen of he me ndex ; l s he oal number of nose varables n he BIW assembly process; η, = 1,,..., L, = 1,,..., m are consans, and, are vecors characerzng he effecs of X( and z, and s a marx characerzng he effecs of he neracon beween X( and z. For BIW assembly processes, z descrbes he random pnhole conac orenaons. he physcal model of a BIW assembly process needs o be suded o oban he coeffcens,,, and η, = 1,,..., L, = 1,,..., m n (. he relaonshp beween process varables and produc ualy characerscs for a sngle fxure saon s gven n Jn and Chen (1 by sudyng he relaonshp among locang pn dameers, par locang errors, and PC devaons. An overall mul-saon process model addressng he varaon propagaon across dfferen saons s reured n he developmen of he process model for a mul-saon BIW assembly process. he sae space model developed n Jn and Sh (1999 for mul-saon shee meal assembly processes wll be used n hs paper o oban he coeffcens of he process 9

10 model. In he followng subsecons, he sae space model and he physcal knowledge of an assembly process wll be revewed o oban he coeffcens of he process model for a BIW assembly process Sae Space Model for Par Locang Errors and PC Devaons From Jn and Sh (1999, he sae euaon a saon can be expressed by: V = H 1V 1( + BF Y = C V =1,,, L (3 where V (, whch s euvalen o he sae vecor n he sae space model, s he par error vecor defned n Jn and Sh (1999 characerzng he dmensonal errors of all ougong pars of saon ; V ( s he dmensonal errors of he raw pars comng from sampng processes; sysem marces H, B, and C encode process confguraon such as he layou of locang ools and PC pons; F ( s he vecor of par locang errors whch s he dmensonal error of he ( Y, 1(! Y, m par a he poson of he locang pns of saon ; and [ ] Y. he frs euaon of he sae space model n E. (3 uses a recursve relaonshp o characerze he propagaon of produc ualy. Based on hs relaonshp, can be seen ha he par dmensonal errors a he curren saon are he accumulaon of par locang errors and raw par dmensonal errors a he prevous saons. 3.. Relaonshp beween Pn Wear and Par Locang Errors he locang pns wear s refleced n he reducon of he pn dameers, whch causes an ncreasng clearance beween a locang pn and he correspondng locang-hole. And hs clearance resuls n he par locang error. he followng noaons are used for he descrpon of relaonshp beween pn wear and par locang errors: (a x P, and z P, denoe he par locang errors of pn P, n he X and Z drecons; 1

11 (b [ ] F ( xp zp x P z,1,1, n P, n! represens he vecor of par locang errors a saon ; (c θ, represens he orenaon of he conacng pon beween he pn P, and he locang-hole. he relaonshp beween he par locang errors ( x, z a four-way locang pn can be obaned as and he pn dameer reducon of P P, ; P =.5X, sn (4 xp =.5X, cosθ,, z θ,, where X, s he accumulaed pn dameer decremen, whch s consdered as he process varable correspondng o he locang pn P, n hs paper. he relaonshp beween he par locang error and he wear of he wo-way locang pn can be obaned as (5 zp =.5X, snθ,, More dealed llusraon of (4 and (5 and more dscusson on dsrbuon of θ are gven n Appendx Process Model for he BIW Assembly Process By recursvely subsung V, V,..., V ( n (3, he PC devaon on he ( 1 1 ougong pars of saon can be calculaed based on boh he par locang errors a saon and hose of prevous saons as below: ( ( Y = G F + ( CH 1H " H V, (6 ( where marx G [( H H " H B ( H H " H B H B B ] and vecor ( F [! ]. ( F1 F F C ! 1 1 z θ! θ, z ( z! ( and, cos,1 snθ,1 cosθ, n sn Le [ ] n, [ ], 1 z, L 11

12 [ V ( ( ] z z. (7 ha s, he raw par error and he random orenaon of he conacng pon beween he pn and he locang-hole are consdered as nose varables n he process model. From (6, (4, and (5, wh some basc algebrac manpulaons, can be seen ha Y (1 ( (1 ( [,, ] + X( [,, ], = 1,,..., L, 1,,..., m = z = = C H H " H ; (1 where [ 1 ] (,: zero marx of approprae dmenson; and (, = where 1 ( ( [ G ] [ G ] # (,1 (( n p 1 k = 1 # (, (( n p 1 z (8 (, s a 1 n vecor whose elemens are all zero; ( ( [ G ](,3 [ G ]( # # (( n p 1,4 (( n p 1!! %!! # # ( ( [ G ] [ G ] (, p 1 (( n p 1 (, p (( n p 1 (1, s a (1 ( n p # (1 ( n p (1 ( n p ( r ( n p p n k. hus, he coeffcens of he process model ( for a BIW assembly process are η, =, = (1, = and [ ] (,,, (1 (, [ ],, 3.3 Produc Qualy Assessmen =. he produc ualy can be assessed by he mean suared devaons of he ualy characerscs he PCs. herefore, he h ualy ndex a he h saon s ( n n, ( X( k = E(( Y, γ, X( k = Var( Y, X( k + ( E( Y, X( k γ,, k < k+ 1 γ s he arge value for Y,. where From he dscusson n subsecon 3. and Appendx 1, can be shown ha E( z =, hus, E( z = and, +, E( z =. hen from (, (, E( Y, X( k γ, = X( k (, +, E( z (, + E( z X( k =,,, and k 1

13 So he locang pn degradaon of a BIW assembly process wll no cause mean shfs of he PCs. he ualy ndex can be wren as a uadrac funcon of X(k as follows, ( X( = Var( Y = X( k k B, k, X( + d, X( = ( X( k k, cov( z (, X( + k, cov( z, (9 where errors, B, =, cov(z, and d =, cov(z,. Snce s only relaed o he raw par d, can be nerpreed as he conrbuon of he raw par errors o he produc ualy a each saon. Le L m E ( ( τ a,, τ = 1 = 1, where a, s he hreshold of he specfcaon for he h PC a he h saon. E represens he even ha no ualy ndex has exceeded s hreshold by me,.e., no falure due o nonconformng producs has occurred by me. 3.4 Relaonshp beween Produc Qualy and Locang Pn Caasrophc Falure Rae he devaon of he locang-hole cener of an ncomng par may accelerae he caasrophc falures of he correspondng locang pn. he larger he varaon of he locanghole poson, he hgher falure chance he pn wll have. he relaonshp beween he caasrophc falure rae λ, ( and he produc ualy ndex ( s descrbed as λ = λ, ( + s, ( X( k where! ( X( k = [ 1,1( X( k 1, ( X( k L, m ( X( L k ] and s, s a vecor of approprae dmenson wh nonnegave elemens and called as he QR-coeffcen. he QRcoeffcen can be calbraed by collecng caasrophc falure daa and produc dmensonal measuremens. he caasrophc falure rae of a four-way locang pn can be affeced by he locanghole varaon n boh he X drecon and he Z drecon. he caasrophc falure rae of a wo- 13

14 way locang pn s affeced only by he locang-hole varaon n he Z drecon snce here s normally lle conac n he X drecon beween a locang-hole and a wo-way locang pn. Based on he relaonshp beween he locang pn caasrophc falure rae and s correspondng locang-hole varaons dscussed above, f P, s a four-way pn, hen f [ ( ] r h r [ s ], = =, f P, s a wo-way pn, hen corresponds o he X or Zdrecon varaon of h locanghole locaed by he pn a he oherwse saon he ; [ s ], = =, f [ ( ] corresponds o he Z drecon varaon of r h r h locang hole locaed by he pn a he oherwse saon he. Here [ ] r denoes he r h elemen of a vecor. 4. Sysem Relably Evaluaon From he defnon of and z n Secon 3..3 and he dsrbuon of θ dscussed n Appendx 1, can be shown ha B = s dagonal. Based on Chen e al.,, cov(z, (1, he produc ualy of such a process does no have self-mprovemen. herefore, followng he same procedures n Chen e al. (1, he sysem relably of a BIW assembly process can be fnally obaned as R( Pr(No pn caasrophc falureand Each ualy ndex s whn specfcaon by c c c = Pr( E AND E = Pr( E Pr( E E = R ( R ( I II (1 where = h s he producon msson me (he relably evaluaon lfeme, R exp( ρ I ( exp( c 1 ~ 1, R df ( x( ~ II (, and E c s defned as he x( X( 14

15 even ha sysem caasrophc falures never occurred by me. Because R s affeced by II ( he ualy consran, whle R s no, can be seen ha R ( = Pr( E R II ( = Pr( E E c ( I. he probables assocaed wh c E and I c, and E are deermned by he pn caasrophc falure rae derved n Secon 3.4 and he ualy ndex derved n 3.3. he dealed c dervaon on how o calculae Pr( E AND E s shown n Chen e al. (1. he parameers c,,, ~ ρ, and dsrbuon of X ( n (1 are gven n Appendx. he followng desgn parameers are used as he npu nformaon o calculae sysem relably for general BIW assembly processes: (a Layou of locang pns (used o ge H, B, and C n he sae space model (b Layou of PCs (used o ge C n he sae space model (c Raw-par error (cov(v ( (d Pn degradaon rae & sandard devaon (µ, (, σ, ( (e Inal pn-hole clearance (µ, (, σ, ( (f Inal caasrophc falure rae (λ, ( (g QR co-effcen s, (h hreshold for ualy ndex a, ( Lengh of me nerval (h In Appendx, he procedure o calculae sysem relably based on he npu nformaon s summarzed n able Opmal Locaor Wear Rae Assgnmen he analycal soluon (1 provdes many poenal applcaons n desgn of a mulsaon BIW assembly process. For example, can be used o deermne he opmal seng of 15

16 he wear raes of locang pns. Suppose ha he wear rae range for each locang pn s defned as µ ( µ µ mn, max < where µ mn and µ max are he lowes and hghes allowable wear raes for each locang pn. And he selecon of he pn wear rae drecly affecs s fabrcaon cos. heorecally, zero s he lower bound for he wear rae. he µ mn n our formulaon s a praccal lower bound. When µ = mn, s euvalen o he problem whou a specfc praccal lower bound. herefore, he opmzaon whou a lower bound (or lower bound s zero s a specal case of our model. We use µ mn because n many suaons f he wear rae s exremely small, may no be achevable based on avalable ool makng echnues. For hese suaons, a posve mnmum value of µ mn wll be more reasonable n he opmzaon problem formulaon. 5.1 Fabrcaon and Coang Coss of Locang Pns ypcally he less wear rae of a locang pn, he more fabrcaon and coang coss wll be suffered assocaed wh he pn. So s reasonable o assume ha he pn wear rae s nversely proporonal o he fabrcaon and coang coss. In hs paper, he coss assocaed wh all locang pns are defned by recprocal funcon of pn wear raes as C p = L n, = 1 = 1 µ, w ( (11 where µ, ( s he wear rae of pn P, and w, s he weghng coeffcen assocaed wh µ, (. 5. Opmzaon Formulaon and Opmaly he obecve of he opmal assgnmen of locang pn wear raes s o maxmze he sysem relably a. he opmzaon problem can be formulaed mahemacally as 16

17 * = Arg max{ R( subec o C p C } max, µ mn µ, ( µ,, max (1 = µ µ 1,1 ( µ 1, (! L, n L ( and max where [ ] fabrcaon cos. C s he budge for he oal oolng Le E denoe he complemen of E. E s he even ha he falure due o nonconformng producs occurs by me. If we frs consran he maxmum wear rae µ max such ha Pr( E < α, he falure due o nonconformng producs can be gnored. he selecon of α depends on specfc manufacurng processes. he BIW assembly process s no a hghly relable process due o s complexy. Normally he sysem relably of neres for a BIW assembly process canno be very hgh and he falure probably of less han.1 due o nonconformng producs s accepable. herefore, for BIW assembly processes, we choose α =.1. I can be seen ha R( = 1 Pr( E c E = 1 Pr( E c Pr( E + Pr( E c E Snce c Pr( E E Pr( E < α, R ( c (1 Pr( E < α herefore, f Pr( E can be gnored, he obecve of maxmzng R becomes maxmzng c c 1 Pr( E = Pr( E = RI ( Now suppose ha he consraned maxmum wear rae µ max s seleced such ha Pr( E µ ( = µ max,, < α (13 he followng resul assures ha under hs condon, he falure due o nonconformng producs can be gnored for all sengs of locang pn wear raes whn he allowable range. ( 17

18 Resul 1. If µ, ( µ, (,,, hen Pr( E (,, ( Pr(, ( = µ = µ E µ µ, ( Proof. hs resul can be easly seen based on he fac ha he produc ualy of he BIW assembly process does no have self-mprovemen. he dealed proof s omed n hs paper. From (13 and Resul 1, he falure due o nonconformng producs can be gnored for any pn wear rae sengs whn he allowable ranges, and he opmzaon problem n (1 s euvalen o he followng opmzaon problem: * = Arg max{ R ( } subec o C p I C max, µ mn µ, ( µ,, max (14 Snce c,, and ~ are ndependen of he decson varable, (14 can be furher rewren as * = Arg mn{ s subec o C p } C max, µ mn µ, ( µ,, max (15 Regardng he opmaly of he above opmzaon formulaon, he followng resuls can be derved based on he form of he analycal soluon. Lemma 1. ρ s a convex funcon of. [ ] Proof. From (19, U ( U ( 1 = + / ( 1 / ρ. Boh U and are symmerc posve semdefne marces based on he dscusson n Chen e al. (1. So he summaon, produc, and nverse of hem are all posve semdefne. herefore, ρ s posve semdefne on. From (19, obvously ρ s also posve semdefne on. Hence s convex on. Lemma. he consran n (15 s a convex se. Proof. Obvously C p s a convex funcon of µ, ( for µ, ( >. So he level se { µ µ } convex and hence ( C, µ µ (, C s a convex se. p max mn, 18 max C p Cmax s

19 From Lemma 1 and Lemma, he resul below abou he opmaly of (15 follows (see Avrel (1976. Resul. he nonlnear opmzaon problem saed n (15 converges o a global mnmum *. 6. Case Sudy A case sudy s conduced o llusrae he developed mehodology. A sde aperure nner panel assembly, as shown n Fgure, s seleced n he sudy. Ral Roof Sde Panel P,3 P,4 &P 3, M 3, M 3,6 M 3,7 P 1, &P, P 1,4 P 3,3 M 3,1 M 3,1 P 1,1 &P 3,1 M 3,3 A-Pllar Inner Panel B-Pllar Inner Panel P 1,3 &P,1 Rear Quarer Inner Panel P 3,4 M 3,4 M 3,5 M 3,9 M 3,8 (a Locang pn posons (b PC posons Fgure. Layou of he oolng posons and PC pons In hs example, four pars are assembled ogeher by hree saons (Fgure 3. A-Pllar and B- Pllar are assembled a saon 1. he subassembly of A-Pllar and B-Pllar s hen assembled wh Ral Roof a saon. A saon 3, he subassembly of A-Pllar, B-Pllar, and Ral Roof are assembled wh he Rear Quarer Inner Panel. he produc ualy s defned by 1 PC pons measured a saon 3 (Fgure (b. he assembly seuences of hese four pars are llusraed n Fgure 3 and he layou of oolng posons s shown n Fgure (a. able 1 and able gve all he dmensons of he oolng posons and he PC pons. he desgn parameers used for hs example are shown n able 3. he raw par dmensonal errors and varaon of nal pn/hole clearance s very small and gnored n hs case sudy. 19

20 P 3,1(4-way P 3,(-way Rear Quarer Inner Panel P 3,3(4-way P 3,4(-way Saon 3 P,1(4-way P,(-way Ral Roof Sde Panel P,3(4-way P,4(-way Saon P 1,1(4-way P 1,(-way A-Pllar Inner Panel B-Pllar Inner Panel P 1,3(4-way P 1,4(-way Saon 1 Locang Pons Fgure 3. Assembly seuences of he sde aperure nner panel able 1. Nomnal X-Z coordnaes for locang pons P 1,1 & P 3,1 P 1, & P, P 1,3 & P 1,4 P,3 P,4 P 3,3 P 3,4 P,1 &P 3, Nomnal X Coordnaes (mm Z able. Nomnal X-Z coordnaes for PC pons PC Pons M 3,1 M 3, M 3,3 M 3,4 M 3,5 M 3,6 M 3,7 M 3,8 M 3,9 M 3,1 Nomnal X Coordnaes (mm Z able 3. Summary of he parameers used n he sudy Descrpon of he parameer Value 7 Inal falure rae λ ( λ ( = λ( = 4 1, s [ ],, QR coeffcen s, = s =.1,,, when he r h elemen r corresponds o he ualy characerscs of he locang hole locaed by he h pn a he h saon Inal pn/hole clearance µ ( µ = µ =.4mm, Operaons per me nerval h Un degradaon rae ( hreshold for ualy ndex a,, (, h = 5 operaons 6 µ ( = ( = 1 mm/opera on, µ, µ, 6 a, = 6a =.8mm,, 5 Un degradaon s..d. σ ( σ ( = ( = 5 1 mm/operaon,, σ

21 In he sudy, wo cases are nvesgaed o demonsrae he conceps and poenal applcaons of he proposed mehodology. (1 Sysem relably analyss wh and whou consderng he QR-Chan effec: he sysem relably analyss s conduced under hree dfferen defnons of sysem falures, whch are: (a Only consder he probably of componen caasrophc falures. ha s, he pn degradaon and he mpac of he ncomng produc ualy on he pn caasrophc falure are no consdered n he model. I s euvalen o he case of seng he QR coeffcen s and he pn degradaon rae µ ( o zero n he QR-Chan model; (b Consder boh he pn caasrophc falure and he produc ualy deeroraon due o componen wear-ou, bu whou consderng he mpac of he ncomng produc ualy on he caasrophc falure rae of he locang pns. I s euvalen o he case of seng he QR coeffcen s n he QR-Chan model o zero; (c Consder he negraed QR-Chan model proposed n hs paper. he sysem relably resuls obaned from he formulae n Secon 4 and he calculaon seps n able 6 of Appendx are shown n Fgure 4. he Malab code for he numercal evaluaon of he sysem relably was run on an IBM PC Penum III machne. In average, akes abou 1 seconds o evaluae sysem relably a a specfc me based on he QR-Chan model. herefore, he evaluaon algorhm used n hs paper s ue effcen and feasble. From he comparson sudy, can be seen ha he sysem relables under defnons (a and (b are always hgher han ha under defnon (c. he overesmaon of (a and (b s no surprsng. Based on he real producon daa, as dscussed n he nroducon, abou 44% of locang ool caasrophc falures n BIW assembly processes are nduced by ncomng produc ualy. As a resul, ncorrec gnorance of he mpac of ncomng produc ualy as n (a and 1

22 (b may lead o sgnfcan overesmaon of he overall sysem relably. If a scheduled manenance polcy s planned based on he predced sysem relably usng defnon (a or (b, many unexpeced down mes could be experenced due o neglecng he nerdependency beween produc ualy and relably of locang pns. Relably 1 (a s=, µ( = (c s, µ( (b s=, µ( Operang Days Fgure 4. Sysem relably wh and whou consderng he QR-Chan effec ( Opmal seng of he wear raes of locang pns: Suppose a preferred prevenve manenance cycle s every 5 producon days based on he producon schedule. From he smulaon resul n Fgure 4, curren sysem relably a he 5 h producon day s only abou.75. In order o reduce he unexpeced sysem falure before he 5 h producon day, he desgner needs o mprove R. he overlap poron of he curves (a and (b n Fgure 4 ( 5 shows ha he sysem falure before he 5 h producon day s manly caused by he sysem caasrophc falure raher han he falure due o nonconformng producs. So he falure due o nonconformng producs by he 5 h producon day can be gnored (wh probably of less han 1%. Furhermore, he dfference beween curve (c and curves (a and (b shows ha he ncomng par ualy has a sgnfcan mpac on he caasrophc falures of locang pns by he 5 h producon day. herefore he desgner can mprove R( 5 by reducng he wear raes of he

23 locang pns so ha he ncomng produc ualy of each saon s mproved and he QR neracons are reduced. In he orgnal case sudy, he wear rae of each locang pn s se o be he same as 6 1 mm/operaon. Assumng w, =w,,, he curren pn fabrcang cos can be calculaed 1w from (11 as C = 6 1. Suppose he desgner wans o mprove R( 5 wh he avalable budge allowng ncremen of he pn fabrcaon cos by a mos 3 1. One smple way o do hs s o reduce he wear rae of each locang pn from 1-6 o mm/operaon. However, he more effcen way s o spend more on reducng wear raes of crcal locang pns deermned by he QR-Chan model. For hs purpose, an opmzaon problem can be formulaed as * = Arg max{ R( 5 } 4 subec o C p C, µ,, 3 6 ( 1, (16 4 hs opmzaon problem s a specal case of (1 wh =5, C max = C, mn 3 µ =, and µ. Also, he aenon can be focused only on he sysem caasrophc falure 6 max = 1 because he falure due o nonconformng producs can be gnored by he 5 h producon day. Based on he dscusson on he opmaly of opmzaon problem (1, he opmal pn wear rae assgnmen s shown n able 4. he opmzaon problem s solved by usng he MALAB funcon fmncon ha uses a Seuenal Quadrac Programmng (SQP mehod (MALAB, he algorhm converges whn 5 seconds, whch s prey effcen. From able 4, can be seen ha here s no need o mprove pns P 1,4, P 3,1, P 3,, P 3,3, and P 3,4 n he opmal soluon. Frs, we already calculaed ha he fnal produc ualy s sasfacory a 5 even f all he locang pn wear raes are no mproved. So leavng hese fve pns no mproved 3

24 wll no resul n poor fnal produc ualy. Secondly, n erms of reducng he pn caasrophc falures before 5 under he consran ha he fnal produc ualy s sasfacory, he obaned resul s conssen wh he physcal undersandng of he process desgn. From Fgures and 3, Pns P 3,1, P 3,, P 3,3 and P 3,4 are used a saon 3, whch s consdered as he fnal saon n hs example whose oupu s he fnal produc raher han he ncomng par of he nex saon. So he degradaon of hese four pns wll no conrbue o he pn caasrophc falures of laer saons. Locang pn P 1,4 conrbues o he roaon movemen of he B-Pllar around he fourway pn P 1,3. hs movemen does no affec he poson of locang-hole for P 1,3, whch s also used as he locang-hole for P,1 a saon. So he degradaon of P 1,4 does no conrbue o he pn caasrophc falure of laer saons eher. Dfferen degrees of mprovemens are performed for oher locang pns based on her geomercal relaonshp and he par locang mechansm. able 4. Opmal pn wear raes * based on he QR-Chan model (1-6 mm/operaon µ 1,1 ( µ 1, ( µ 1,3 ( µ 1,4 ( µ,1 ( µ, ( µ.3 ( µ,4 ( µ 3,1 ( µ 3, ( µ 3,3 ( µ 3,4 ( able 5 s used o compare he orgnal sysem falure probably a 5 (whch s 1-R( 5, he mproved falure probably based on unform mm/operaon wear raes for each pns, and he mproved falure probably based on he opmal soluon *. Seng he wear raes unformly mproves he sysem falure probably by 9.7%. By usng he opmal wear rae seng based on he QR-Chan model, he sysem falure probably can be mproved by 3.%, wh he pn fabrcaon cos remanng he same as ha of he unform wear rae seng. herefore, wh he ad of he QR-Chan model developed n hs paper, opmal desgn can be 4

25 acheved o maxmze he sysem relably (or mnmze sysem falure probably under he consrans of avalable budge on oolng fabrcaon coss. able 5. Comparson of 1-R( 5 for hree dfferen sengs of pn wear raes 1-6 mm/operaon mm/operaon * F( 5 =1-R( Summary Qualy and relably are wo mporan facors n manufacurng sysem desgn. In BIW assembly processes, real producon daa have shown sgnfcan neracons beween locang ool relably and produc ualy. hs paper developed a process model of a mul-saon BIW assembly process based on he process knowledge. he QR-Chan model s appled based on he process model o capure he QR neracon and s propagaon hrough mul-saon BIW assembly processes. Process and produc desgn nformaon of assembly sysems were negraed no he QR-Chan model and a sae space model was employed o sudy he varaon propagaon hrough all assembly saons. An analycal soluon for sysem relably evaluaon has been obaned based on he QR-Chan model. A case sudy s conduced n hs paper, whch shows ha he sysem relably wll be overesmaed f he QR neracon s mproperly gnored. An opmzaon problem, as a ypcal applcaon of he QR-Chan model, s formulaed o opmally assgn pn wear raes wh consraned budge on pn fabrcaon coss. he opmaly of he opmal soluon s derved based on he analycal form of he sysem relably. he opmal desgn obaned from he QR- Chan model sgnfcanly mproves he sysem relably compared wh ha obaned from curren pracce. I should be noced ha fuure research s sll needed on how o selec and desgn echnues o acheve he assgned opmal wear rae for each locang pn. Anoher neresng 5

26 fuure sudy followng hs paper s he negraon of he manenance decson-makng and he wear rae desgn for locang pns, hrough whch an overall opmal producvy of BIW assembly processes can be acheved. Acknowledgemen In hs research, Dr. Y. Chen s suppored by he General Moor Collaborave Research Lab a he Unversy of Mchgan; Dr. J. Sh s parally suppored by he Naonal Scence Foundaon (NSF Gran: NSF-DMI ; and Dr. J. Jn s suppored by he Naonal Scence Foundaon (NSF Career Gran: CAREER AWARD NSF-DMI 13394, and Socey of Manufacurng Engneers Educaon Foundaon Gran: Research Inaon Award 5. References Apley, D. and Sh, J. (1998 Dagnoss of Mulple Fxure Fauls for Panel Assembly. ASME ransacons, Journal of Manufacurng Scence and echnology, 1, Avrel, M. (1976 Nonlnear Programmng. Prence-Hall, Inc., New Jersey. Ceglarek, D. and Sh, J. (1995 Dmensonal Varaon Reducon for Auomove Body Assembly Manufacurng. Journal of Manufacurng Revew, 8, Ceglarek, D. and Sh, J. (1996 Fxure Falure Dagnoss for Auo Body Assembly Usng Paern Recognon. ASME ransacons, Journal of Manufacurng Scence and echnology, 118, Ceglarek, D., Sh, J. and Wu, S. M. (1994 A nowledge-based Dagnosc Approach for he Launch of he Auo-Body Assembly Process. ASME Journal of Engneerng for Indusry, 116, Chen, Y., Jn, J. and Sh, J. (1 Relably Modelng and Analyss of Mul-saon Manufacurng Processes Consderng he Qualy and Relably Ineracon. Proceedngs of IEEE Inernaonal Conference on Sysems, Man, and Cybernecs, ucson, AZ. Dng, Y., Ceglarek, D. and Sh, J. ( Modelng and Dagnoss of Mulsage Manufacurng Process: Par I Sae Space Model. Proceedngs of he Japan-USA Symposum on Flexble Auomaon. Dng, Y., Jn, J., Ceglarek, D. and Sh, J. (3a Process-Orened olerancng for Mul-Saon Assembly Sysems. Acceped by IIE ransacons. 6

27 Dng, Y., m, P., Ceglarek, D. and Jn, J. (3b Opmal Sraegy of Sensor Dsrbuon for Dagnosng Varaon Sources n Mul-saon Assembly Processes. Acceped by IEEE ransacons on Robocs and Auomaon. Dng, Y., Sh, J. and Ceglarek, D. ( Dagnosably Analyss of Mul-saon Manufacurng Processes. ASME Journal of Dynamc Sysems, Measuremen, and Conrol, 14, Hu, S. and Wu, S. M. (199 Idenfyng Roo Causes of Varaon n Auomoble Body Assembly Usng Prncpal Componen Analyss. ransacon of NAMRI, XX, Jn, J. and Chen, Y. (1 Qualy and Relably Informaon Inegraon for Desgn Evaluaon of Fxure Sysem Relably. Qualy and Relably Engneerng Inernaonal, 17, Jn, J. and Sh, J. (1999 Sae Space Modelng of Shee Meal Assembly for Dmensonal Conrol. ASME ransacons, Journal of Manufacurng Scence and echnology, 11(4, MALAB (1999 Opmzaon oolbox User s Gude, Verson 5. he MahWorks Inc., Nack, MA. Shu, B., Ceglarek, D. and Sh, J. (1997 Flexble Beam-based Modelng of Shee Meal Assembly for Dmensonal Conrol. NAMRI/SME ransacons, 4, Yang, S., Chen, Y. and Lee, J. ( Modelng of Assembly Lnes and Fxures for Varaon and Relably, General Moors Saelle Research Laboraory. Zhou, S., Dng, Y., Chen, Y. and Sh, J. (3 Dagnosably Sudy of Mulsage Manufacurng Processes Based on Lnear Mxed-Effecs Models. Acceped by echnomercs. Appendx Appendx 1. Deals on Relaonshp beween Pn Wear and Par Locang Errors he relaonshp beween pn wear and he par locang error n a fxure saon has been suded n Jn and Chen (1. Based on observaons from lab and real auobody assembly plans, n mos cases he locang pn ouches he wall of he locang-hole durng he assembly operaons. So s reasonable o assume ha he locang-hole conacs wh he pn on one sde when he par s posoned by a fxure. Due o he sll exsng possbly ha he locang pn does no ouch he locang-hole, hs assumpon may resul n slgh overesmaon of he produc varaon and conservave predcon of he relably. 7

28 From he assumpon above, he conacng orenaons beween he locang pn and he locang-hole for a four-way pn and a wo-way pn are shown n Fgure 5. he par locang error n he X-Z plane can be represened by he dsplacemen of he locang-hole cener from he cener of he pn as shown n Fgure 5. Based on Fgure 5, he relaonshp beween he par locang errors ( x, z and he pn dameer reducon of a four-way pn can be obaned as P P, x, P X, z P,,, z P Z P, (4-way X, P, (-way O X Fgure 5. Par locang error due o pn wear ; P =.5X, sn (17 xp =.5X, cosθ,, z θ,, Here θ, are assumed o be ndependen random varables followng unform dsrbuon,, whn [, π], whch s denoed as θ, ~U(, π. I can also be obaned ha Var(sn θ = Var(cosθ, =.5 and Cov(sn θ,cosθ, = for a four-way locang pn. Smlarly, gnorng he effec of he wear of a wo-way pn n he X drecon, he relaonshp beween he par locang error and he wear of he wo-way pn can be obaned as (18 zp =.5X, snθ,, Because he locang-hole conacs wh he wo-way pn eher on he upper or he lower sde n he Z drecon, θ, s a random varable havng wo values of -π/ and π/ wh he same 8

29 probably eual o.5. In hs paper, s denoed as θ, ~Unf {-π/, π/} for wo-way locang pns. I can also be obaned ha Var(sn θ, = 1 for a wo-way locang pn. Appendx. Parameers n (1 and procedure o calculae sysem relably Based on he relably resuls for a general manufacurng process wh QR-Chan effec n Chen e al. (1, he parameers n (1 s gven as ( c L n = 1 = 1 L n λ ( + s d, where = 1 = 1, d [ d 1,1 d1, d L, m d L 1 L, m ], L! d, s obaned n (9; ( ( (1, # (, (,1 (1 # (,1!! %! (, (1, # ( ( n( + 1 n( + 1 From (1, ( k +, k cov( X ( +, X( = ( k, ( k, k + = ( k +, k = ( k, and k k ( k = ( k 1 + Q, k = 1,,..., ( = ; ( s a doman n n R s.. L m x ( { x ( B x( a, d, } ; ~ ~ ~ (v he dsrbuon of X ( s X ( ~ N( ~ (, (, where ~ ~ ( and ( can be = 1 obaned by paronng a marx ~ and a vecor ~ as ~ = 11 1 ( n n ( n n 1 ( n n ( n n, ~ ( n 1 =, and ~ ~ ( = ( and =. From Chen e ( n 1 al. (1, ~ ~ and can be calculaed by 1 1 ( + / ( / 1 ~ 1 U, U + ; ~ 1 9

30 where I = U ( L n 1 mk k ( n n U, U h [ ] ( n n s, B + l k, l and 1 k 1 m k 1 ( n n = 1 = 1 k= 1 l= 1 = 1 ; (v ρ can be calculaed as 1 1 [ U ( U + ( / ( / ] > ρ =, (19 ( (1! (, ( k = ( k 1 + ε, k 1, and ( =. where [ ] he general procedure o calculaed sysem relably based on he npu nformaon lsed n Secon 4 s summarzed n he followng able. able 6. General procedures o calculae sysem relably of BIW assembly processes Sep Oupus Inpus Relaed formula/resuls n he paper 1 cov(z (c * (7 and Appendx 1 (a and (b Secon 3..3 and (3 3 Q and 6 (d and (e (1 4 (d and (e (19 5 d (a, (b, and cov(z from Sep 1 (8 and (9 6 B from sep and cov(z from sep 1 (9 7 c (f, (g, and d from sep 5 Appendx ( 8 6 Q and 6 from sep 3 Appendx ( 9 (h, d from sep 5, and B from sep 6 Appendx ( 1 U (g, (, B from sep 6 Appendx (v 11 ρ from sep 4, 6 from sep 8, and Appendx (v 1 U from sep 1 ~ from sep 4, U from sep 1, and 6 from sep 8 Appendx (v 13 6 from sep 8 and U from sep 1 Appendx (v 14 ~ F~ X ( I ( ~ from sep 13 and ~ from sep 1 Appendx (v ρ from sep 11, 6 (1 ~ from sep 13 F~ X from sep 14 (1 15 R c from sep 7, from sep 8, and 16 R II ( ( 17 R( R I ( from sep 15 and R II ( from (1 sep 16 * he npu numberng n parenhess follows ha of he npu ls n Secon 4. 3

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon