Simulated Analysis for Multivariate GR&R study

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1 Smulatd Analy for Multvarat GR&R tudy Rogéro S. Pruch *, Andron P. Pava *, Pdro P. Baltra *, Albrto Garca-Daz + * Inttut of Indutral Engnrng and Managmnt, Fdral Unvrty of Itajubá, Itajubá, MG, Brazl + Dpartmnt of Indutral & Informaton Engnrng, Unvrty of nn, Knoxvll, N, USA Emal: rogropruch@unf.du.br, andronppava@unf.du.br, pdro@unf.du.br, agd@utk.du Abtract h artcl xplor th analy of maurmnt ytm wth corrlatd charactrtc through th tudy of rpatablty and rproducblty. h man contrbuton of th rarch to propo a mthod for multvarat analy of a maurmnt ytm by condrng th wghtd prncpal componnt (WPC). o prov t ffcncy, w gnrat mulatd data wth dffrnt corrlaton tructur for maurmnt ytm that ar accptabl, margnal, and unaccptabl. h propod mthod compard wth clacal unvarat and multvarat mthod. It wa obrvd that, compard to th othr mthod, th WPC wa mor robut n tmatng th amnt ndc of a multvarat maurmnt ytm. Kyword: maurmnt ytm analy, rpatablty and rproducblty, prncpal componnt analy, corrlatd ualty charactrtc. Introducton Qualty mprovmnt projct ar oftn charactrzd by thr objctv to rduc varablty and achv zro-dfct producton. If a product fal to conform, analyt gnrally attrbut th falur to th proc. h analyt thn act to mprov proc capablty. In om ntanc, howvr, thr may b nothng wrong wth th proc capablty. Yt th maurmnt rror, whn compard to th varablty of th proc, rman unaccptabl (Al-Rfa & Bata, 00). Hnc, bfor a tam of analyt tr to mprov a proc, thy hould nvtgat both th varablty of th maurmnt proc a wll a th varablty of th manufacturng proc. In manufacturng, a maurmnt ytm not ud to producng th xact dmnon of a part. It provd maurmnt that, du to rror (random and ytmatc), vary from th tru valu (AIAG, 00). In any actvty nvolvng maurmnt, om of th obrvd varablty du to th product tlf, P, whl th rmandr du to maurmnt rror or varablty n th maurmnt ytm, (L & Al-Rfa, 008; Snol, 00; Woodall & Borror, 008). In maurmnt ytm analy (A), th tudy ud to maur th componnt of varaton calld Gag Rpatablty and Rproducblty (GR&R). Rpatablty th varaton n maurmnt obtand wth on maurng ntrumnt whn ud vral tm by an apprar whl maurng th dntcal charactrtc on th am part. Rproducblty th varaton n th avrag of maurmnt mad by dffrnt apprar ung th am gag whn maurng a charactrtc on on part (Awad t al., 009; Burdck t al., 003; Erdmann t al., 00; Poln & urchtta, 00). GR&R am to dtrmn that a maurmnt ytm varablty l than that of th montord proc (Al-Rfa & Bata, 00). In ang maurmnt ytm that maur multpl charactrtc, unvarat approach may prov unatfactory, a mght th tratgy of prortzng th CQ. In uch ytm, an analy mut condr th corrlaton tructur of th CQ, a tak mor utd to multvarat mthod. For GR&R tud, th ltratur rval fw multvarat tud (Wang & Chn, 00). h artcl dal wth a multvarat analy of a maurmnt ytm through tud of rpatablty and rproducblty of th maurmnt proc. It man contrbuton to propo a nw mthod for multvarat analy of a maurmnt ytm bad on prncpal componnt analy. h nw mthod Wghtd Prncpal Componnt (WPC) pondr th prncpal componnt cor by thr gnvalu. o prov th ffcncy ID3.

2 ICIEOM 0 - Gumarã, Portugal of th mthod, mulatd data ar gnratd wth dffrnt corrlaton tructur o a to maur ytm that ar unaccptabl, margnal, and accptabl. h rult obtand by WPC wll b compard to tho obtand by Prncpal Componnt Analy (PCA) and Multvarat Analy of Varanc (MANOVA) mthod. h mulaton tudy conclud that th propod mthod mor robut than MANOVA (Majk, 008) and PCA (Wang & Chn, 00). h rmandr of th papr tructurd a follow. Scton how a brf rvw about how to obtan prncpal componnt cor and how to valuat a maurmnt ytm ung th WPC mthod propod by th author. In Scton 3, a mulaton tudy conductd to valuat th prformanc of th multvarat mthod for dffrnt corrlaton tructur and for unaccptabl, margnal, and accptabl maurmnt ytm. Fnally, Scton prnt th man fndng nvolvng th analy ung th multvarat mthod MANOVA, PCA, and WPC. Multvarat GR&R tudy bad on PCA and WPC Accordng to Wang & Chn (00), to dal wth multpl CQ n a GR&R tudy, PCA an altrnatv mthod to th MANOVA propod by Majk (008). Prncpal componnt analy on of th mot wdly appld tool ud to ummarz th common pattrn of varaton among varabl. Furthrmor, th tattcal tchnu alo abl to rtan gnfcant nformaton n th frt ax of th PC, nc th varaton aocatd wth xprmntal rror, maurmnt rror, roundng rror ummarzd n th lat ax of PC (Pava t al., 007). Prncpal componnt ar algbracally a lnar combnaton l of random varabl CQ,CQ,, CQ. Gomtrcally th combnaton rprnt a nw coordnat ytm obtand durng th rotaton of an orgnal ytm (Johnon & Wchrn, 00; Pava t al., 008). h coordnat of th ax now hav th varabl CQ,CQ,, CQ and rprnt th drcton of maxmum. h prncpal componnt ar uncorrlatd and dpnd only on th varanc-covaranc matrx Σ (or th corrlaton matrx R) of varabl CQ,CQ,, CQ and thr dvlopmnt do not rur th aumpton of multvarat normalty. Snc th matrx ha par of gnvalu-gnvctor λ λ λ, thn th th prncpal componnt : λ,, λ,,, λ,, whr 0 PC Y Y Y Y,,, () h th prncpal componnt can b obtand accordng to: Maxmz : Subjct to : Var CQ Cov CQ, CQ 0, k An orgnal t of varabl can b rplacd by an uncorrlatd lnar combnaton calld prncpal componnt cor. Condrng Z, th matrx of tandardzd data, and E, th matrx of gnvctor of th multvarat t, ach prncpal componnt cor can thn b obtand from (H t al., 0; Johnon & Wchrn, 00): k () PC cor Z y y E y n y y y y y y n y y y y y y n y y y (3) ID3.

3 Smulatd Analy for Multvarat GR&R tudy E. () rprnt a complt modl for a multvarat GR&R tudy wth ualty charactrtc, p part, o oprator, and r rplcat that can b analyzd by PCA. h modl mlar to th unvarat modl. h orgnal rpon, howvr, ar rplacd by th prncpal componnt cor. PC μ α β j αβ ε,,, p; j,,, o; k,,..., r j jk h varabl μ a contant and α, β j, (αβ) j, ε jk ar ndpndnt normal random varabl wth zro man and varanc,,,, and, for part (proc), oprator, part * oprator ntracton, and rror trm, rpctvly. In thr analy of maurmnt ytm, Wang and Chn (00) compard th PCA wth two othr mthod. Howvr, th author conductd th analy paratly for ach prncpal componnt. h mthodology may b napproprat bcau th ndvdual analy of ach componnt can provd dffrnt ntrprtaton. Whn rpon hav vry hgh corrlaton (%PC >95%), analy of th frt prncpal componnt xplan raonably wll th varablty of th maurmnt ytm. Whn corrlaton btwn th rpon ar not vry hgh, howvr, t bcom ncary to analyz mor than on prncpal componnt. Indd, th frt prncpal componnt alon cannot xplan th whol data t. hrfor, w propo n th artcl, a mthod of a multvarat GR&R tudy ung wghtd prncpal componnt. In th ca, th rpon of th modl prncpal componnt cor wghtd by thr rpctv gnvalu. h propoal bad on th work of Pava t al. (00), who ud a tchnu of mult-objctv optmzaton bad on a wghtng of th prncpal componnt. hy ud th tchnu to tudy a proc of wldng wth a multpl t of rpon modratly corrlatd. hrfor, th propod modl gvn by: whr: WPC μ α β j αβ ε,,, p; j,,, o; k,,..., r j WPC jk λ PC λ PC λ PC λ PC () (5) (6) hat, th rpon ud n modl (5) th rult of a pondrng of th prncpal componnt by thr gnvalu, accordng to E. (6). h varabl μ a contant and α, β j, (αβ) j, ε jk ar ndpndnt normal random varabl wth zro man and varanc,,,,, rpctvly. In Johnon and Wchrn (00), t appar that thr ar a varty of rul to tmatng th approprat numbr of non-trval PCA ax (PC cor) that can b takn to rprnt th orgnal data t. Howvr, du to th wghtng of th prncpal componnt by thr gnvalu, all prncpal componnt can b ncludd n th modl. h componnt wth gnvalu of gratr mportanc wll, n th modl, b wghtd mor, and whol nformaton wll b ncludd n th tudy. h componnt of varanc n modl (5) can b tranlatd nto GR&R notaton by: P PO ˆ ˆ P (7) or ˆ ˆ E (8) rpatablty O PO PO E σˆ σˆ σ ˆ rproducblty β αβ (9) pr r ˆ ˆ ˆ (0) rpatablty rproducblty ˆ () ˆ ˆ P P, O, PO, and E ar, rpctvly, th man uar for th factor part, oprator, ntracton, and th rror trm. If th ntracton ffct not gnfcant, th complt modl can b rducd to E. () and t componnt of varanc can b tranlatd nto a GR&R notaton by E. (3)-(7). WPC μ α β ε,,, p; j,,, o; k,,..., r () j jk P E σˆ P (3) or ID3.3

4 ICIEOM 0 - Gumarã, Portugal σˆ σˆ E () rpatablty ε O E ˆ ˆ rproducblty (5) pr ˆ ˆ ˆ (6) rpatablty ˆ ˆ P rproducblty ˆ (7) h amnt ndx of maurmnt ytm ud to compar th prformanc of th mthod %R&R (rpatablty and rproducblty prcntag) n E. (8). h accptanc crtron for th maurmnt ytm of th multvarat %R&R m ndx th am for th unvarat %R&R ndx (Majk, 008; Wang & Chn, 00). If %R&R m l than 0%, th maurmnt ytm condrd accptabl. If %R&R m btwn 0% and 30%, th maurmnt ytm condrd margnal accptabl dpndng on th applcaton, th cot of th maurmnt dvc, th cot of rpar and othr factor. If, accordng to th ndx, th maurmnt ytm xcd 30%, thn t condrd unaccptabl and hould b mprovd (AIAG, 00; Al-Rfa & Bata, 00; Burdck t al., 003; Woodall & Borror, 008). % R & R m 00% (8) 3 Smulaton h purpo of th mulaton to valuat almot all pobl tuaton n multvarat analy of a maurmnt ytm and to compar th rult achvd through multvarat mthod. Smulatd data wll b gnratd for maurmnt ytm that ar unaccptabl (%R&R>30%), margnal (0%<%R&R<30%), and accptabl (%R&R<0%), a wll a corrlaton that ar vry low (%PC 65%), low (65%<%PC 75%), mdum (75%<%PC 85%), hgh (85%<%PC 95%), and vry hgh (%PC >95%), a total of 5 cnaro and 800 mulatd maurmnt. %PC th rult obtand from. Fgur xmplf two xtrm ca, vry hgh corrlaton (Fgur a) and vry low (Fgur b) btwn two rpon varabl CQ and CQ. It obrvd that th gratr th mlarty pattrn of chang of factor lvl (part and oprator), th gratr th corrlaton btwn th charactrtc. Alo, f you t vry dffrnt man valu for two oprator maurng th am part, th analy of varanc ndcat tattcally gnfcant dffrnc btwn oprator and/or ntracton trm (part*oprator). Fgur how a flowchart that dtal how to obtan th mulatd data. Accordng to th flowchart, th mulatd data wr gnratd from th nformaton n abl, accordng to th am amount of part and oprator n Majk (008), p=5, o= and r=3. h data for th 5 mulatd cnaro can b found n h mulaton tudy wll focu only on th comparon of multvarat mthod. o undrtand how to calculat unvarat %R&R ndx ung ANOVA mthod, AIAG (00). h calculaton of multvarat %R&R m ndx through MANOVA and PCA mthod can b found n Majk (008) and Wang and Chn (00), rpctvly. hu, th %R&R m and %R&R ndc wr calculatd to compar th multvarat mthod. For ach cnaro w trd to obtan clo %R&R ndx valu for CQ, CQ, CQ 3, and CQ. It xpctd that th ndc obtand by multvarat mthod ar clo to tho obtand by ANOVA mthod. h crtron ud n th work to dtrmn f th tmatd multvarat ndx, %R&R m, corrct bad on confdnc ntrval for man calculatd from data obtand by ANOVA mthod. h lowr (LLCI) and uppr (ULCI) lmt of th confdnc ntrval ar calculatd ung E. (9) and (0): LLCI CQ t N, N (9) ULCI CQ t N, N (0) whr CQ th man of %R&R btwn CQ, CQ, CQ 3, and CQ ; th tandard dvaton; N th th ampl z and t N, th ( )00 prcntl of a t dtrbuton wth (N-) dgr of frdom. ID3.

5 Smulatd Analy for Multvarat GR&R tudy (a) Sampl Man 6 0 CQ M CQ M 6 _ UCL=0.89 X=8.6 0 LCL= _ UCL=.53 X=9.36 LCL=6.9 (b) Sampl Man Part 3 5 _ UCL=.89 X=9.6 LCL= Part 3 5 _ UCL=.53 X=9.36 LCL=6.9 Fgur : Xbar chart by oprator: (a) xampl of a vry hgh corrlaton (0.999) btwn th rpon CQ and CQ ; (b) xampl of a vry low corrlaton (0.088) btwn th rpon CQ and CQ SAR. Crat a gnral varanccovaranc matrx. Crat man vctor for ach oprator (o) maurng ach part (p) 3. Gnrat r (rplcat) data for ach combnaton o maurng p. Stack combnaton to form th matrx of rpon 6.3. Idntfy hgh corrlaton to modfy man vctor 5. Analyz th prncpal componnt of th rpon 6.. Analyz th corrlaton btwn th rpon 6.5. Idntfy low corrlaton to modfy man vctor 6. I %PC th rurd valu? 6.. I mallr %PC rurd? 6.. Analyz th corrlaton btwn th rpon 7. Analyz th maurmnt ytm 8. I %R&R th rurd valu? 8.. Dcra varanccovaranc matrx by multplyng a calar 8.. I mallr %R&R rurd? 8.3. Incra varanccovaranc matrx by multplyng a calar END Fgur : Dtald flowchart xplanng how to obtan mulatd data to a multvarat GR&R tudy ID3.5

6 ICIEOM 0 - Gumarã, Portugal abl : Man vctor and varanc-covaranc matrc ud to gnrat mulatd data wth dffrnt corrlaton and maurmnt ytm () Scnaro Vry Low corr. Unaccptabl Low corr. Unaccptabl 3 Mdum corr. Unaccptabl Hgh corr. Unaccptabl 5 Vry hgh corr. Unaccptabl 6 Vry Low corr. Margnal 7 Low corr. Margnal 8 Mdum corr. Margnal 9 Hgh corr. Margnal 0 Vry hgh corr. Margnal Vry Low corr. Accptabl Low corr. Accptabl 3 Mdum corr. Accptabl Hgh corr. Accptabl 5 Vry hgh corr. Accptabl Man vctor P O P O P 3 O P O P 5 O P O P O P 3 O P O P 5 O ,00,00 8,00 0,00,00 6,0,0 7,99 9,99,0 3,00 6,00 9,00,00 5,00 3,0 6,0 9,0 0,99,99 6,00 8,00,00 5,00 3,00 6,0 8,0,0 5,0 3,0 8,00 0,00,00 6,00,00 7,99 0,0,0 6,0, ,00,00 8,00 0,00,00 6,0,0 7,99 9,99,0 3,00 6,00 9,00,00 5,00 3,0 6,0 9,0 0,99,99 6,00 8,00,00 5,00 3,00 6,0 8,0,0 5,0 3,0 8,00 0,00,00 6,00,00 7,99 0,0,0 6,0, ,00,00 8,00 0,00,00 6,0,0 7,99 9,99,0 3,00 6,00 9,00,00 5,00 3,0 6,0 9,0 0,99,99 6,00 8,00,00 5,00 3,00 6,0 8,0,0 5,0 3,0 8,00 0,00,00 6,00,00 7,99 0,0,0 6,0, Varanc-covaranc matrx ID3.6

7 Smulatd Analy for Multvarat GR&R tudy abl prnt th rult of calculaton of th %R&R ndx a wll a th man valu and th 95% confdnc ntrval, obtand by ANOVA mthod. abl 3 how th rult of calculaton of th man valu, 95% confdnc ntrval and %R&R m ndx, obtand by PCA, MANOVA and WPC mthod. h analy and comparon wll b prformd n two way: ntra- and ntr-mthod. h ntra-mthod analy wll provd an ovrvw of th mthod prformanc to tmat th %R&R m ndx. h ntrmthod analy wll k to jutfy th mthod dvaton of tmat of th %R&R m ndx from th confdnc ntrval. abl : Rult for calculaton of th %R&R ndx, man and 95% confdnc ntrval. SCENARIO UNIVARIAE (%R&R) MEAN CI S Corrlaton CQ CQ CQ 3 CQ Man LLCI ULCI S Vry Low S Low S3 Unaccptabl Mdum S Hgh S5 Vry Hgh S6 Vry Low S7 Low S8 Margnal Mdum S9 Hgh S0 Vry Hgh S Vry Low S Low S3 Accptabl Mdum S Hgh S5 Vry Hgh abl 3: Rult for calculaton of th man, 95% confdnc ntrval and %R&R m. MEAN CI MULIVARIAE (%R&R m ) S Man LLCI ULCI PC PC PC 3 PC MANOVA WPC S (55.8) 9.55 (9.) 5.3 (.).08 (.0) S (70.7) 0.8 (7.5) 0.9 (8.) 9.3 (3.7) S (79.). (9.) 7.65 (7.5) 9.6 (3.7) S (88.3) 8.9 (8.) 9.6 (3.) 75.9 (0.) S (99.7) (0.) (0.0) (0.0) S (.9) 6.9 (9.9).79 (3.5) 3.3 (.7) S (66.).77 (7.8) 8.37 (5.9) 6.89 (0.) S (79.8) 6.0 (9.).78 (7.5) 7.9 (3.5) S (89.8) 6.0 (7.6) 9.99 (.5) 5.07 (0.).3.00 S (99.9) (0.) (0.0) (0.0) S (5.5) 6.39 (35.9).59 (7.) 5.83 (.).08.0 S (67.6).5 (8.3).5 (3.8) 5.33 (0.) S (79.7).95 (8.9) 5.75 (7.8) 3.08 (3.6) S (90.3) 3.3 (7.) 3.96 (.6) 33. (0.) S (00.0) (0.0) (0.0) (0.0) In th ntra-mthod analy vrfd that, n th tmaton of th %R&R m ndx, th WPC wa mor robut than th MANOVA and PCA. h MANOVA mthod wa abl to tmat th multvarat ndx wthn th confdnc ntrval only n cnaro S9, S and S. For PCA mthod, Wang and Chn (00) valuatd th prncpal componnt that had a cumulatv prcntag of xplanaton at lat 95% for th orgnal varabl. hu, th PCA mthod wa capabl to tmat th multvarat ndx wthn th confdnc ntrval only n cnaro S5, S0, S and S5. A n n abl 0, WPC tmatd %R&R m ndx wthn th confdnc ntrval for all 5 cnaro valuatd. For th ntr-mthod analy by PCA, abl 3 prnt th analy of maurmnt ytm mulatd to th four prncpal componnt. h valu n parnth how th xplanaton prcntag of varablty ID3.7

8 ICIEOM 0 - Gumarã, Portugal of th CQ for ach prncpal componnt. It rcommndd valuat th cor of prncpal componnt that rprnt at lat 95% of cumulatv varablty for CQ. hrfor, cnaro wth corrlaton tructur: Vry low, low and mdum: PC, PC and PC 3 wr analyzd; Hgh: PC and PC wr analyzd; Vry hgh: only PC wa analyzd. In S5, S0 and S5, only PC wa analyzd, th rult howd that PCA wa capabl to tmat %R&R m wthn th confdnc ntrval. For othr cnaro, PC wa nuffcnt to provd a raonabl xplanaton of varablty of th CQ. hu, whn othr prncpal componnt wr analyzd, %R&R m ndx wa tmatd outd th confdnc ntrval (xcpt for S). In hort, whn th corrlaton tructur btwn th CQ rur that othr prncpal componnt ar analyzd, bd PC, th PCA mthod may fal ( abl 3). For th ntr-mthod analy by MANOVA, abl prnt how th %R&R m ndx wa tmatd for th 5 mulatd cnaro. It vrf that th mthod wa capabl to tmat th multvarat ndx wthn of th confdnc ntrval only n S9, S and S. h ndx wa obtand by MANOVA ung gomtrc man of accordng to th amount of ualty charactrtc. h mulaton tudy dalt wth four charactrtc. hrfor, four gnvalu of th ˆ and ˆ matrc wr xtractd. If th ndvdual rlatonhp for ach par of gnvalu,,, 3 and, n ˆ and ˆ, provd dffrnt ntrprtaton, th %R&R m ndx tmatd by MANOVA may not rprnt wll th prformanc of th maurmnt ytm. h du to th fact that th gomtrc man provd th am dgr of mportanc n th analy of ach par of gnvalu. Howvr, t known that th frt gnvalu hav a gratr prcntag of xplanng th maurd phnomnon gratr than th lat gnvalu. hrfor, th nd confrmd that om form of wghtng for th calculaton of th ndx hould b ud. abl : %R&R m ndx for th ntr-mthod analy by MANOVA S Man LLCI ULCI 3 3 S (6.) 3.9 (7.) 5.3 (0.5) 6.3 (0.9) 0.78 S (7.6) 9.35 (3.9) 8.3 (7.7) 8.76 (3.9) 3.30 S (80.) 5.8 (0.8) 7.80 (5.) 7.87 (3.6).3 S (87.3) 5.70 (8.6).66 (3.) 8.78 (.) 8.5 S (99.8) 5.77 (0.) (0.0) 9.3 (0.0) 6.09 S (53.). (9.0).0 (6.3) 7.3 (.3).97 S (67.7). (8.3) 3.37 (3.9).73 (0.) 0.0 S (8.) 3.00 (.0).5 (.6) 3.5 (3.) 5.0 S (90.). (6.9) 0.03 (.6) 0.7 (0.).3 S (99.9) (0.0) 70.5 (0.0) (0.0) 7.3 S (50.).80 (3.5). (3.5) 6. (.).08 S (7.5) 0.95 (5.5) 0.73 (.7) 3.80 (0.3).0 S (8.5).3 (0.6).69 (.6).68 (3.3).8 S (90.8) 7.39 (6.).0 (.8) 3.63 (0.) 7. S (00.0) 38.9 (0.0) 98. (0.0) 8.0 (0.0) In th ntr-mthod analy by WPC, abl 3 how that th %R&R m ndx wa tmatd wthn th confdnc ntrval for th 5 mulatd cnaro. WPC wa mor robut than PCA and MANOVA bcau t ovrcam om hortcomng of th mthod. For PCA, whn PC nuffcnt to xplan th whol varablty of CQ, othr prncpal componnt can provd valuaton for th maurmnt ytm outd th confdnc ntrval. MANOVA provd a gnral ntrprtaton for th maurmnt ytm, howvr th tratgy of ung gomtrc man wa not atfactory. In WPC, th tratgy of ID3.8

9 Smulatd Analy for Multvarat GR&R tudy wghtng th cor of prncpal componnt by thr gnvalu provd to b uffcnt to corrct th hortcomng prvouly mntond. Morovr, th mulaton tudy howd that th hghr corrlaton btwn th CQ, th clor to th man valu wll b th tmat of %R&R m ung th WPC mthod. Concluon h artcl addrd th multvarat analy of maurmnt ytm through tud of rpatablty and rproducblty of th maurmnt proc. h man contrbuton of th papr t propoal for a nw mthod for multvarat analy of th maurmnt ytm by wghtng th prncpal componnt. o prov th ffcncy of th mthod, mulatd data wr gnratd wth dffrnt corrlaton tructur for maurmnt ytm condrd accptabl, margnal, and unaccptabl. h rult obtand by WPC mthod wr compard to tho obtand by th multvarat mthod (PCA and MANOVA). Morovr, tattcal analy provdd th followng concluon: PCA mthod may fal whn corrlaton tructur btwn CQ not condrd uffcntly hgh and mor than th frt prncpal componnt rurd to b analyzd; MANOVA mthod u gomtrc man to tmat multvarat ndx for valuatng th maurmnt ytm. h approach may b ncorrct whn th rlaton for ach par of gnvalu provd gnfcant dffrnc for thr calculaton; akng all tuaton of th mulaton tudy nto account, WPC mthod howd mor robut than PCA and MANOVA mthod. WPC wa abl to ovrcom hortcomng uch a: to provd an ngl amnt for all CQ n multvarat GR&R tudy; to tmat th multvarat %R&R m ndx nd th confdnc ntrval vn whn th corrlaton tructur of CQ condrd low; and to provd a tratgy of wghtng that guarant gratr mportanc for prncpal componnt mot tattcally gnfcant to tmat th %R&R m ndx. 5 Acknowldgmnt h author would lk to thank FAPEMIG, CAPES and CNP for thr upport n th rarch. Rfrnc AIAG. (00). Maurmnt Sytm Analy: Rfrnc Manual, fourth d. Automotv Indutry Acton Group, Dtrot, MI. Al-Rfa, A., & Bata, N. (00). Evaluatng maurmnt and proc capablt by GR&R wth four ualty maur. Maurmnt, 3, Awad, M., Erdmann,.P., Shanhal, Y., & Barth, B. (009). A maurmnt ytm analy approach for hard-to-rpat vnt. Qualty Engnrng,, Burdck, R.K., Borror, C.M., & Montgomry, D.C., 003. A rvw of mthod for maurmnt ytm capablty analy. Journal of Qualty chnology, 35, Djaghr, B., Jmdar, M., D Smt, M., Cockart, P., Smyr-Vrbk, J., & Vandr Hydn, Y., 006. Improvng mthod capablty of a drug ubtanc HPLC aay. Journal of Pharmacutcal and Bomdcal Analy, Dldo, L., & Zappa, D., 0. Maurmnt uncrtanty wth ntd mxd ffct modl. Qualty and Rlablty Engnrng Intrnatonal 7, Erdmann,.P., Do, R.J.M.M., & Bgaard, S., 00. Qualty uandar: a gag R&R tudy n a hoptal. Qualty Engnrng, H, S.G., Wang, G.A., & Cook, D.F., 0. Multvarat maurmnt ytm analy n multt ttng: An onln tchnu ung prncpal componnt analy. Exprt Sytm wth Applcaton 38, Johnon, R.A., & Wchrn, D., 00. Appld Multvarat Stattcal Analy. ffth d. Prntc-Hall, Nw Jry. ID3.9

10 ICIEOM 0 - Gumarã, Portugal Kaja, K., Pkkann, V., Mäntyalo, M., Koknn, S., Nttynn, J., Halonn, E., & Mankkamäk, P., 00. Inkjttng dlctrc layr for lctronc applcaton. Mcrolctronc Engnrng 87, L, M.H.C., & Al-Rfa, A., 008. Improvng woodn part ualty by adoptng DMAIC procdur. Qualty and Rlablty Engnrng Intrnatonal, Lyu, J., & Chn, M.N., 008. Gaug capablty tud for attrbut data, Qualty and Rlablty Engnrng Intrnatonal, 7-8. Majk, K.D., 008. Approval crtra for multvarat maurmnt ytm. Journal of Qualty chnology 0, 0-5. Pava, A.P., Cota, S.C., Pava, E.J., Baltra, P.P., & Frrra, J.R., 00. Mult-objctv optmzaton of puld ga mtal arc wldng proc bad on wghtd prncpal componnt cor. Intrnatonal Journal of Advancd Manufacturng chnology 50, 3-5. Pava, A.P., Frrra, J.R., & Baltra, P.P., 007. A multvarat hybrd approach appld to AISI 500 hardnd tl turnng optmzaton. Journal of Matral Procng chnology 89, Pava, A.P., Pava, E.J., Frrra, J.R., Baltra, P.P., & Cota, S.C., 008. A multvarat man uar rror optmzaton of AISI hardnd tl turnng. Intrnatonal Journal of Advancd Manufacturng chnology 3, Poln, & W., urchtta, S., 00. t protocol for mcro-gomtrc war of ntrd damond tool. War 57, Snol, S., 00. Maurmnt ytm analy ung dgnd xprmnt wth mnmum α-β Rk and n. Maurmnt 36, 3-. Wang, F.K., & Chn,.W., 00. Proc-orntd ba rprntaton for a multvarat gaug tudy. Computr and Indutral Engnrng 58, Wang, F.K., & Yang, C.W., 007. Applyng prncpal componnt analy to a GR&R tudy. Journal of th Chn Inttut of Indutral Engnrng, Woodall, W.H., & Borror, C.M., 008. Som rlatonhp btwn gag R&R crtra. Qualty and Rlablty Engnrng Intrnatonal, ID3.0

Outline. Types of Experimental Designs. Terminology. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Lecture 12

Outline. Types of Experimental Designs. Terminology. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Lecture 12 EEC 686/785 Modlng & Prformanc Evaluaton of Computr Sytm Lctur Outln Rvw of lctur r Factoral Dgn wth Rplcaton Dpartmnt of Elctrcal and Computr Engnrng Clvland Stat Unvrty wnbng@.org (bad on Dr. Ra Jan