A Simple Procedure to Calculate the Control Limit of Z Chart

Transcription

1 Inrnaional Journal of Saisics and Applicaions 214, 4(6): DOI: /j.saisics A Simpl Procdur o Calcula h Conrol Limi of Z Char R. C. Loni 1, N. A. S. Sampaio 2, J. W. J. Silva 2,3,*, R. B. Ribiro 3 1 Univrsidad Esadual Paulisa, UNESP, Guarainguá, SP, Brazil 2 Associação Educacional Dom Bosco, AEDB, Rsnd, RJ, Brazil 3 Faculdads Ingradas Trsa D Ávila, FATEA, Lorna, SP, Brazil Absrac A Z charcan b usd o monior procss qualiy characrisics. Whn hr is corrlaion bwn obsrvaions of wo masurabl qualiy characrisics, X and Y, and hr is dpndnc on h im among obsrvaions of X and also Y and his srucur of corrlaion and auocorrlaion is of a VAR(1) modl, i is possibl, for a crain fals alarm ra, o rla h conrol limi of h Z char wih h variancs and covariancs of h cross-covarianc mari. This papr proposs a linar rgrssion modl o drmin h conrol limi of Z char. Th mhod found in liraur for obaining h conrol limi of h Z char is lowr han h linar rgrssion modl proposd in his aricl; i is mor complicad and dos no guaran h dsird fals alarm ra. Kywords Z conrol char, Linar rgrssion, Auocorrlaion 1. Inroducion Th saisical procss conrol hlps managrs undrsand, monior and coninuously improv h qualiy of producs and srvics. In h 193s, Shwhar [1] crad h conrol chars o monior procsss, and hn rcognizd h nd o monior procsss considring h mulivaria conrol. Th radiional conrol chars assum by mans of hypohsis indpndnc among obsrvaions of variabl ha on wishs o monior. Howvr, high producion spds gnra corrlaion among h qualiy characrisics and dpndnc among obsrvaions of on qualiy characrisic of nighbor producs according o h manufacuring insan [2]. Som sudis wr don in ordr o valua h chars prformanc of mulivaria conrol in auocorrlaion prsnc, concluding ha hr is a drop in hs chars prformanc [3-6]. Th mulivaria procsss monioring whos obsrvaions ar auocorrlad appars in rcn publicaions. Masranglo and Forrs [7] hav mad availabl a program o gnra auocorrlad daa whr i is possibl o simula displacmn in valu of avrag of variabl undr monioring. Pan and Jarr [8] proposd h us of was of h VAR(p) modl o monior auocorrlad procsss. Th chniqu rquirs fiing h modl o procss daa for lar us of was in h T 2 char. Arka [5] maks us of arificial nural nworks for monioring * Corrsponding auhor: jwjsilva@gmail.com (J. W. J. Silva) Publishd onlin a hp://journal.sapub.org/saisics Copyrigh 214 Scinific & Acadmic Publishing. All Righs Rsrvd mulivaria auocorrlad procsss. Issam and Mohamad [6] propos h us of h SVR (suppor vcor rgrssion) mhod o monior changs in h man vcor in auocorrlad procsss hrough h MCUSUM conrol char. Hwarng and Wang [9] sablishd h us of nural nworks ha ar abl o idnify shifs in h man vcor of auocorrlad procsss. Thr ar svral ohr works on monioring auocorrlad procsss [1-13]. Auocorrlaion compromiss h us of conrol char, for fals alarms incras whn i is disrgardd, ha is, whn h conrol limis ar sablishd undr hypohsis of auocorrlaion absnc [7, 14-16]. Kalgonda and Kulkarni [3] proposd h Z char o monior wo or mor qualiy characrisics for commns which follow a VAR(1) modl. Th advanag of Z char in rlaion o T 2 char is ha i idnifis h qualiy characrisic ha suffrs chang in is avrag valu. Th auhors prsn an mpirical procdur o drmin h conrol limi (CL) of Z char. Thy assum ha hr is a corrlaion bwn h obsrvaions of X and Y and hr is dpndncy in h im bwn X and Y obsrvaions and his corrlaion and auocorrlaion srucur is of a VAR(1) modl. This aricl shows ha for his corrlaion and auocorrlaion srucur hr is a linar rlaionship bwn h CL of Z char and variancs and mari covarianc of cross-covarianc of X and Y. For a wid rang of valus of cross-corrlaion and auocorrlaions, i was obaind is a cofficin for drmining h linar rlaionship modl highr han.95. This aricl aims o prsn a linar rgrssion modl o obain h conrol limi of h Z char ha nsurs h fals alarm ra dsird for a wid rang of valus of cross

2 Inrnaional Journal of Saisics and Applicaions 214, 4(6): corrlaions and auocorrlaions. Th only mhod dscribd in h liraur for obaining h CL was h Kalgonda and Kulkarni mhod [3] which, bsids complicad, almos always givs conrol limis mor spacd han i is ncssary o m h dsird fals alarm ra. Th papr is organizd as follows: i is prsnd, in scion 2, h modl ha dscribs h qualiy characrisics of a procss wih cross-corrlaions and auocorrlaions; in scion 3, h Z conrol char; scion 4 prsns h rgrssion modl for obaining h CL of Z char and compars h fals alarm ra calculad using h limis obaind by rgrssion wih h fals alarm ra calculad using h limis obaind by h mhod of Kalgonda and Kulkarni [3]. 2. Modl Dscribing h Qualiy Characrisics Th classical conrol procdurs in mulivaria procsss considr h basic assumpion ha obsrvaions follow a mulivaria normal disribuion and ar indpndn wih vcor of mans µ and h varianc-covarianc mari Σ. X = µ + = 1, 2,..., T (1) whr X rprsns obsrvaions by a vcor of ordr pppp1 (pis h numbr of variabls); ar indpndn random vcors of ordr pppp1 wih mulivaria normal disribuion whos man is zro and varianc-covarianc mari Σ. Th indpndnc assumpion is violad in many manufacuring procsss, which maks quaion (1) inadqua o rprsn such obsrvaions. Vcor of auo rgrssion of firs-ordr, or VAR(1), quaion (2), hav bn usd o modl mulivaria procsss wih mporal corrlaion among obsrvaions of a variabl and corrlaion among obsrvaions of diffrn qualiy characrisics [2-6, 9, 17-2]. In mulivaria auocorrlad procsss, h VAR(1) modl is rprsnd as follows: X µ =Φ( X µ ) + (2) 1 whr X is h daa vcor of ordr pppp1; µ is h man vcor of ordr pppp1 and Φ is a mari conaining auorgrssiv paramrs of ordr p p and ar indpndn random vcors of ordr pppp1 wih mulivaria normal disribuion whos man is zro and varianc-covarianc mari Σ. If Φ is a zro mari, quaion (2) is rducd o quaion (1), ha is, on has h classical modl for indpndn daa ovr h im. Ohrwis, h daa will b dpndn ovr h im and h modl variaion srucur is rprsnd by h cross-covarianc mari givn by quaion (3) [21]. Undr h assumpion ha h procss is saionary EX ( ) µ =, for all, cross-covarianc mari will b: ( µ )( µ ) E X X h =Γ ( h) h=,1,2,... Bing saionary mans ha µ is consan for all X and h cross-covarianc mari dos no dpnd on, i dpnds only on h which rprsns h inrval ovr h im and bwn h vcor X and X. Th mari ) givn by: h (3) Γ (h is formd by h lmns γ (h) γ ij ( h) = E ( Xi µ )( X j h µ ) i, j = 1, 2,..., p Sinc h cross-covarianc mari originally dpnds on h masurmn uni of involvd variabls, somims is inrpraion is no simpl. A mor convnin way o valua h rlaionship of variabls in h procss is givn by using h cross-corrlaion mari: ij (4) ρ ( h) = D Γ ( hd ) (5) whr D is h diagonal mari formd by h lmns γ ij (h), for all i=j, of mari Γ (h). Th cross-covarianc mari for h=, Γ (), whn Φ and ar known, can b obaind by h raio of Yul-Walkr [24]: ' Γ ( ) = ΦΓ () Φ + (6) Assuming X is a daa vcor wih p-varid disribuion and follow h modl dscribd in quaion (2), according o Kalgonda and Kulkarni [3] and Kalgonda [2], [ ] X ~ N µ ; Γ () (7) p If h procss is in saisical conrol, X follows a mulivaria normal disribuion wih man vcor and cross-covarianc mari Γ (). 3. Z Conrol Char Wih h simulanous us of X chars o conrol wo or mor qualiy characrisics, i is possibl o idnify which of hm has bn affcd by h spcial caus. Howvr, whn h variabls ar dpndn or corrlad, o obain h conrol limis of h X chars is no longr rivial [22], for h probabiliy ha h valus of X 1, X 2,..., X p, ar wihin h conrol limis is no mor givn by: ( 1 α ) p (8) whr p is h numbr of variabls and α h probabiliy of a fals alarm.

3 278 R. C. Loni al.: A Simpl Procdur o Calcula h Conrol Limi of Z Char Kalgonda and Kulkarni [3] proposd a conrol char calld a Z char for monioring h man vcor of mulivaria auocorrlad procdurs. Th char mainains h ovrall rror α and allows h variabls idnificaion whos mans hav changd wih h mrgnc of a spcial caus. Th auhors adapd h saisical conrol chniqu of mans vcor for indpndn obsrvaions proposd by Hayr and Tsui [25] and considrd ha auocorrlaion in h procss follows h VAR(1) modl. A im insan, h Z monioring saisics of Z char is givn by Z = Ma1 i p Z i, whr: X µ = ; = 1,2,..., ; = 1, 2,... (9) γ i i Zi i p ii,() whr X i is h valu of h i-h variabl a insan of im and γ ii,() is h i-h diagonal lmn of h cross covarianc mari o h=. For a crainα valu, h CL of h Z char is givn by: Pr Z LC; i = 1, 2,..., p µ = µ = 1 α i i (1) Th procss is considrd in saisical conrol if Z LC. Ohrwis, hr is vidnc ha h man of a las on of h p variabls changd. Th disribuion of Z saisic is no known; Kalgonda and Kulkarni [3] obaind h CL by simulaion following h sps: Sp 1. Gnraing a larg numbr of vcors (N = 1) wih obsrvaions according o h sandard p-varia ( ; p X ) modl X N µ ρ ( ) ; Sp 2: Calculaing h Z saisic for ach of h N vcors gnrad in sp 1; Sp 3 Obaining h mpirical disribuion of h Z saisic, find h sparari of ordr (1 α ) and assign his valu o h CL. Th sps dscribd by Kalgonda and Kulkarni [3] almos always lad o conrol limis mor widly spacd han ncssary o m h dsird ra of fals alarms (ARLo> 1 α ). Th ARLo is h avrag numbr of obsrvaions among fals alarms. For indpndn and uncorrlad variabls h ARLo = 1 α [23]. To illusra, l i b h bivaria cas (p=2): µ = (,) ; Σ = and Φ=.5 1,.7 hn from (6) i has bn obaind h cross-covarianc mari, γ () = γ () =.984 (11) Γ () = γ21() =.984 γ22() = Th mhod proposd by Kalgonda and Kulkarni [3] provids for α =.5 a CL of For CL = 3.191, i has bn obaind by simulaion on ARLo = Th appndi provids dails of simulaion. Bcaus of auocorrlaion ARLo dos no follow a gomrical disribuion, for h probabiliy α of fals alarm is no consan. Dpnding on h paramrs of h VAR(1) Modl, h CL of Z char providd by h mhod of Kalgonda and Kulkarni [3] lads o diffrn ARLos. In ordr o solv his problm, his papr proposs a linar rgrssion modl which provids h CL of Z char corrsponding o h dsird ARLo (s Figur 1 of scion 4). 4. Proposd Mhod In ordr o facilia h us of h Z char, h CL valus wr obaind by simulaion for a wid rang of paramr valus of auocorrlaion mari and of h covarianc mari of bivaria VAR(1) rror. Two rgrssion modls wr mad, on for ARLo of 2 and anohr for ARLo of 37. In rgrssion modls simaion, h CL valus wr allocad o h dpndn variabl vcor and h lmns valus of cross-covarianc mari wr allocad o h indpndn vcors mari. Th modl fid o h daa providing R 2 valus vry clos o 1, s Tabls 2 and 4. Modl paramrs for ARLo of 2 and 37 ar shown in Tabls 1 and 3, rspcivly. Tabl 1. Paramrs of rgrssion modl - ARLo = 2 Cofficin Sandard rror raio- p-valu Consan <.1 γ () <.1 11 γ () <.1 22 γ () <.1 12 Tabl 2. Saisics of h modl in Tabl 1 Saisics Valu Sum of squard rsiduals.1 R-squar.99 Saisical F (3.98) Sandard rgrssion rror.3 Adjusd R-squard.99 P-valu (F). Wih h valus of γ () 11, γ () 22 and γ () 12 of cross-covarianc mari i is possibl o obain h conrol limis of h Z char.

4 Inrnaional Journal of Saisics and Applicaions 214, 4(6): For ARLo=2: CL= γ 11 () γ 22 () - For ARLo=37: γ 12 () (12) CL= γ 11 () γ 22 () γ 12 () (13) I has also bn considrd h cas whr h ARLo is qual o 37. Th rsuls of rgrssion modl ar shown in Tabls 3 and 4. Tabl 3. Rgrssion modl paramrs - ARLo = 37 Cofficin Sandard Error raio- p-valu Consan <.1 γ () <.1 11 γ () <.1 22 γ () <.1 12 Tabl 4. Saisics of modl in Tabl 3 Saisics Valu Sum of squard rsiduals.169 R-squard.9466 Saisics F(3. 98) Sandard rror of rgrssion.116 Adjusd R-squard.9453 P-valu(F) Snsiiviy Analysis of h Proposd Mhod To illusra h us of h proposd mhod and is diagnosicabiliy in h prsnc of auocorrlaion, on considrs a similar cas of bivaria vcor as prsnd in Kalgonda and Kulkarni [3]. Th rsuls in Figur 1 illusra h abiliy of h proposd mhod o valua h CL. On considrs h following scnarios o carry ou h analysis: ARLo is qual o 2, i has bn adopd valus a and b a of mari Φ = ranging from.2 o.8 and valus b 1 ρ ρ of h mari Σ = ρ 1 qual o.3,.5 and.7. In pracic hs valus ar unknown; h lmns of h cross-covarianc mari ha dpnd on a, b and ρ ar simad according o quaion (6). I is imporan o noic ha an inrsing subjc of rsarch is h sudy of Z char in mulivaria procss. Figur 1. NMAF = ARLo obaind by rgrssion and Kalgonda mhod [2]

5 28 R. C. Loni al.: A Simpl Procdur o Calcula h Conrol Limi of Z Char Th ARLo valus for 48 diffrn scnarios ar prsnd in Tabl B1 of Appndi B and wr usd in h consrucion of Figur 1. From Tabl B1 and Figur 1, i is obsrvd ha h rgrssion modl in ordr o obain h CL is br han h Kalgonda and Kulkarni mhod [3], for i kps h ARLo always clos o 2 for all scnarios. 5. Conclusions This papr has prsnd a mhod br han h on proposd by Kalgonda and Kulkarni [3] for obaining h CL of Z char. Br in ordr o provid conrol limis ha lad o fals alarms ras closr o hos dsird. Th mhod of Kalgonda and Kulkarni [3] provids gnrally CL valus largr han h on ncssary; his cssiv procion agains fals alarms occurrnc rducs h conrol char abiliy o dc changs in h procss. Th mhod proposd in his aricl rquirs gra ffor for h consrucion of h linar rgrssion modl; howvr, afr obaining i, h calculaion of CL of Z char is immdia. Appndi A Mhod Usd in Simulaion of Mulivaria Tmporal Sris wih Gnraion Procss VAR(1) Simulaion of a mulivaria mporal sris wih p dimnsion and T siz: 1) I is crad rrors wih Gaussian mulivaria ~ N ; Σ, by mans of disribuion of ordr p, ( ) p muliplicaion of mari P of ordr (p p) wih vcor V = (v 1,..., v p ) of ordr (p1), whr PP = Σ and V ~ N(,1). v 1 = P v p (A1) For insanc, if p=2: 1 p11 v1 = (A2) 2 p21 p22 v2 1) Th sp 1 is rpad T ims for gnraion of a sor of rrors. 2) Wih valus, i is obaind X in a rcursiv way by quaion A3 urning =1,2...,T. X µ =Φ( X µ ) + (A3) 1 whr: X is a mari of ordr( p 1); µ is a man mari of ordr(p 1); Φ is h auocorrlaion mari of ordr(p p). 3) Wih vcor gnrad in (3), i is obaind h saisics Z = Ma1 i p Z i. 4) Th CL of Z char is calculad by a binary sarch unil h ARLo is qual o h dsird valu. Appndi B ARLo valus Scnarios a b ρ Tabl B1. Comparison of ARLo valus basd on rgrssion modl and on Kalgonda and Kulkarni mhod [3] γ () γ () γ () CL Rgrssion ARLo* CL Kalgonda ARLo**

6 Inrnaional Journal of Saisics and Applicaions 214, 4(6): * ARLo obaind wih h CL of rgrssion mhod (12). ** ARLo obaind wih CL of mhod proposd by Kalgonda and Kulkarni [3]. procsss: Daa and shif gnraion, Journal of Qualiy Tchnology, Vol. 34, N o 2, p , 22. REFERENCES [1] Shwhar, W. A. Economic conrol of qualiy of manufacurd produc. 1ª Ed. Nw York: D. Van Nosrand Company [2] Kim, S. B.; Jipiaklr, W.; Sukchora, T. On-Class Classificaion-Basd Conrol chars for Monioring Auocorrlad Mulivaria Procsss. Communicaions in Saisics - Simulaion and Compuaion, v.39, n.3, p , 21. [3] Kalgonda, A. A.; Kulkarni, S. R. Mulivaria qualiy conrol char for auocorrlad procsss. Journal of Applid Saisics, v.31, p , 24. [4] Jarr, J. E; Pan, X. Th qualiy conrol char for monioring mulivaria auocorrlad procsss. Compuaional Saisics & Daa Analysis, v.51, p , 27. [5] Arka, J.; Niaki, S. T. A.; Abbasi, B. Arificial nural nworks in applying MCUSUM rsiduals chars for AR(1) procsss. Applid Mahmaics and Compuaion, v. 189, p , 27. [6] Issam, B. K.; Mohamad, L. Suppor vcor rgrssion basd rsidual MCUSUM conrol char for auocorrlad procss. Applid Mahmaics and Compuaion, v.21, p , 28. [7] Masranglo, C.; Forrs, D. Mulivaria auocorrlad [8] Pan, X; Jarr, J. E. Using vcor auorgrssiv rsiduals o monior mulivaria procsss in h prsnc of srial corrlaion. Inrnaional Journal of Producion Economics, v.16, p , 27. [9] Hwarng, H. B.; Wang, Y. Shif dcion a sourc idnificaion in mulivaria auocorrlad procss. Inrnaional Journal of Producion Rsarch, v.48, n.3, p , 21. [1] Aply, D.W.; Tsung, F. Th auorgrssiv T 2 char for monioring univaria auocorrlad procsss. Journal of Qualiy Tchnology, v.34, p.8-96, 22. [11] Jiang, W. Mulivaria conrol chars for monioring auocorrlad procsss. Journal of Qualiy Tchnology, v.36, p , 24. [12] Vargas, M; Alfaro, J. L.; Mondéjar, J. On h run lngh of a sa-spac conrol char for mulivaria auocorrlad procss daa. Communicaions in Saisics - Simulaion and Compuaion, v.38, p , 29. [13] Chn, S; Nmbhard, H. B. Mulivaria cuscor conrol chars for monioring h man vcor in auocorrlad procss. IIE Transacions, v.43, p , 211. [14] Harris, T. J.; Ross, W. H. Saisical procss conrol procdurs for corrlad obsrvaions. Canadian Journal of Chmical Enginring, v.69, p.48-57, [15] Woodall, W. H.; Falin, F. W. Auocorrlad daa and SPC.

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