A Comparison on the MRL Performances of Optimal MEWMA and Optimal MCUSUM Control Charts

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1 IC 11 Hol quaoral Bag-Purajaya, Malaysa, Jauary 11 L P U T PR J O R TI D I OCI O I UIV UDT T R I TI I K B DO G I M L Y I l a o a r I f C o IC 11 f r 1 Prodg of h Iraoal Cofr o dvad, grg ad Iformao Thology 11 Hol quaoral Bag-Purajaya, Malaysa, Jauary 11 ICIT 11 Prodg of h Iraoal Cofr o dvad, grg ad Iformao Thology Cug dg s for Fuur usaably 1 IB Orgazd by Idosa uds ssoao Uvrs Kbagsaa Malaysa Comparso o h MRL Prformas of Opmal MWM ad Opmal Corol Chars Mhal Boo Chog Khoo*, Y Th, z Kog Chuah, Tg Fug Foo, hool of Mahmaal s, Uvrs as Malaysa 118 Md, Pag, Malaysa Tl.:+(6) , -mal: mkb@usm.my* bsra Th MWM (alld h mulvara xpoally wghd movg avrag) har ad h (alld h mulvara umulav sum) har ar usd pross moorg wh a quk do of small or modra shfs h ma vor s dsrd. Th prmary objv of hs sudy s o ompar h prformas of h opmal MWM ad opmal hars basd o hr mda ru lgh (MRL) profls. Th umbr of qualy hararss osdrd s p =. Two ass ar sudd,.., Cas 1 (a shf oly o varabl) ad Cas (a shf wo varabls). Mo Carlo smulao s odud usg asal alyss ofwar () o sudy ad ompar h MRL prformas for varous maguds of ma shfs wh h pross s ormally dsrbud. Ovrall, h rsuls show ha h MRL prformas of h MWM ad hars ar omparabl. Kywords Mda ru lgh (MRL), MWM, I. ITRODUCTIO May smoduor maufaurrs ad hmal pross plas maa maufaurg daabass o hudrds of varabls. Of h oal sz of hs daabass s masurd mllos of dvdual rords. Th moorg of hs daa wh uvara sasal orol hars s of msladg. Th us of mulvara mhods has rasd graly r yars for hs raso, as mod [1]. Hollg [] frs rodud h Hollg s T mulvara qualy orol har for moorg h pross ma vor. h, h applao of h Hollg s T orol har as a pross moorg ool has bam rasgly popular h fld of sasal pross orol (PC). I s ommo o moor svral rlad varabls or qualy hararss smulaously wh h roduo of modr daaaquso qupms ad ompurs (.g. [3]). Th Hollg s T orol har whh a b usd boh Phas I ad Phas II suaos s a hwhar-yp orol har. I uss formao oly from h urr sampl ad gors ay formao gv by h r squ of sampls. s a rsul, s ssv o small ad modra shfs h ma vor. Two ohr yps of mulvara orol hars whh s basd o a Phas II produr,.., h mulvara xpoally wghd movg avrag (MWM) ad mulvara umulav sum () hars, ar proposd as supror alravs o h Hollg s T har, whr hy ak o aou of h prs ad pas formao abou h pross. Opmal sasal dsgs of h MWM ad hars, basd o avrag ru lgh (RL) ad mda ru lgh (MRL) hav b proposd, as [4] ad [5]. Th MRL, whh s h 5 h prag po of h ru lgh dsrbuo s suggsd o b usd as a poal alrav o h RL. Ths s du o h fa ha h orol ru lgh dsrbuos of h MWM ad hars ar hghly skwd, h rprao basd o h RL a b msladg. I addo, h skwss of h ru lgh dsrbuo hags aordg o h magud of h shf h ma vor ad hs maks rprao basd o RL mor omplx. Thus, MRL s usd o valua h prformas of orol hars hs sudy. boh h MWM ad hars hav omparabl RL prformas, as [6], w ar rsd o ompar h prformas of h opmal MWM ad opmal hars, basd o hr MRL profls, for moorg of h ma vor of a mulvara ormally dsrbud pross. ff orol har s abl o d saaously a pross shf ad adops h ssal orrv aos o mprov h pross qualy. Th MRLs ar obad usg a Mo Carlo smulao. Th ma vor s allowd o hag for varous szs of shfs, so ha h prformas of h hars, basd o MRL a b ompard. 31

2 Th rmadr of hs papr s orgazd as follows: os II, III, IV ad V rvw h MWM har, opmal dsg of h MWM har, har ad opmal dsg of h har, rspvly. smulao sudy s arrd ou o ompar h MRL prformas of h opmal MWM ad hars o VI. Fally, som usful olusos ar summarzd o VII. II. TH MWM COTROL CHRT Th MWM sass proposd [7] s dfd as follows: Ζ = rx + ( 1 r) Z, = 1,,, (1) whr Z = s h al vor of h MWM sas ad r ( < r 1) s h smoohg osa. L X 1, X,, b h dpdly ad dally dsrbud mulvara ormal radom vors, ah wh p ompos. I s assumd whou loss of graly ha h o-arg pross ma vor, μ s a vor of zros. Th plod valus o h MWM har ar T = Z Σ z Z, = 1,,, () r whr 1 ( 1 r) r = Z X s h ovara marx of Z ad X s h ovara marx of X. Th MWM sas rdus o h Hollg s T sas f r = 1. Th MWM orol har gvs a ou-of-orol sgal as soo as T > H, = 1,,, (3) whr h orol lm H > s a osa ha s hos o ahv h dsrd -orol MRL (MRL ). III. TH OPTIML DIG OF MWM CHRT MWM har s opmal dg a shf f has h smalls ou-of-orol MRL (MRL 1 ) amog all MWM hars wh h sam MRL for h sam magud of shf. Th opmal dsg of a MWM har spfs h opmal slo of h osas, r ad s orrspodg orol lm, H. abl provdg h opmal ombaos of r ad H for sld umbr of varabls, -orol RLs ad szs of shfs, for h opmal MWM har s gv [8]. h opmal ombaos of r ad H gv [8] ar lmd, hs opmal ombaos ar xrad from h graphs gv [4]. Du o spa osra, h graphs [4] ar o show hs papr. Irsd radrs a rqus hs papr from s sod auhor. IV. TH COTROL CHRT vral rsarhrs hav dvlopd mulvara xsos of h uvara CUUM har (.g. [9], [1], [11], [1], [13], [14]). Two hars ar proposd [9], whr h o wh h br RL prforma s basd o h followg sass: ad ( X a) ( X a) C X = + +, = 1,,..., (4), C k = k ( + X a ) 1, C > k (5) C whr a s h am po or arg valu for h ma. I hs papr, s assumd whou loss of graly ha a =. o ha = ad k > s h rfr valu of h shm. ordg o [9], a ou-of-orol sgal s grad wh Y = > H, (6) X whr H > s h orol lm. Th produr s of basd o h assumpo ha h obsrvao X, for = 1,,, p, blogs o a dpdly ad dally dsrbud pross, from a mulvara ormal dsrbuo. V. TH OPTIML DIG OF CHRT opmal har s dfd as h har ha has h smalls ou-of-orol MRL (MRL 1 ) a a parular shf h pross ma, amog all h hars wh h sam MRL. Th opmal paramr k of h har, drmd basd o RL as a rro o b mmzd, s approxmaly gv as half of h sz of h shf,, whr s h oraly paramr gv as follows [9]: = ( μ μ) ( ), X μ μ (7) Ths valu of k appars o mmz h RL 1 for a parular magud of shf, basd o a gv RL. fr obag h opmal rfr valu of k, o ds o oba h orrspodg orol lm, H. Th opmal paramrs, k ad H for a opmal har, basd o h dsrd magud of shf for a quk do ad MRL, ar ak from [5]. o ha h abls gvg h opmal paramrs ar o dsplayd hr baus of spa lmao. Howvr, hs papr oag h abls a b rqusd from s sod auhor. VI. COMPRIO OF MRL PRFORMC: MWM VRU CHRT Th ma am of PC hqus s o d as arly as possbl h prs of assgabl auss of varao ha aff h qualy of a pross. Ths sudy ompars h opmal MWM ad opmal hars for moorg h ma vor by assssg how hr MRLs dffr. Wh h pross s -orol, h MRL should b suffly larg o mmz fals alarms. Wh h pross s ou-of-orol, h har should hav a small MRL, so ha pross shfs a b dd promply. Th prformas of h MWM ad hars ar ompard by sg MRL = 37 for boh hars ad ompar hr MRL 1 s, for a gv shf h pross. Th maguds of shfs h 3

3 ma vor ar fxd a {,.5,.1,.15,.,.5,.3,.35,.4,.45,.5,.75, 1,, 4, 6, 8, 1, 1, 18}. Th umbr of qualy hararss osdrd s p=. Two ass ar osdrd hs papr, whr Cas 1 volvs a shf of oly o varabl whl Cas volvs a shf of wo varabls. Th ou-of-orol ma vors for Cass 1 ad ar rspvly, μ s = (δ, ) ad μ s = (δ, δ). Th valus of MRL 1 for h MWM ad hars ar ompud va smulao usg h program. Hr, MRL 1 s dfd as h mda umbr of bvara obsrvaos ha mus b plod o h MWM (or ) har bfor h har sgals a ou-of-orol, wh a pross shf ours. To xpla how MRL 1 valus ar ompud, h MWM har s osdrd ad xplad as follows: Frs, a squ of bvara ormal obsrvaos ar grad for Cass 1 ad, followd by ompug h MWM sass usg h formula quaos (1) ad (). Wh h MWM har ssus a ou-of-orol ( T > H, for = 1,, ), say a =, h valu of s rordd as h ru lgh valu orrspodg o h ral. Ths pross s rpad for 5 rals, whr a ru lgh valu s ompud for ah ral. Th h mda of h 5 ru lgh valus, obad from h 5 rals s ak as h MRL 1 valu. Th sam produr s usd o oba h MRL 1 valus for h har. Tabl I shows h MRL 1 s for h MWM ad hars. For h MWM har wh MRL = 37 ad a shf of =.1, for whh a quk do s dsrd, w oba r =.8 ad H = 5.6 from h graphs gv [4]. mlarly, for h har wh MRL = 37 ad a shf of =.1, for whh a quk do s dd, w oba k =.9 ad H = 18.5 from h abl gv [5]. Th opmal paramrs Tabls II o IV ar obad usg h sam mhod. Tabls I, II, III ad IV gv MRL 1 rsuls for opmal s of.1,.3,.5 ad 1, rspvly. Th MWM ad hars ar mor ffv dg small shfs. For xampl, osdr h har Cas 1 of Tabl I. Wh rass from.5 o.1, h MRL 1 rdus sgfaly from 39 o 1. Th MRL 1 drass furhr from 1 o 14 wh rass from.1 o.15. Th wh rass from o 1, h har s MRL 1 drass from 1 o,.., a a slowr pa ompard o h spd of dras MRL 1 dsussd abov. smlar rd s also obsrvd for h MWM har. Thus, h ffvss of h MWM ad hars h do of shfs ar mor prooud wh s small. x, w ompar h prformas of h opmal MWM ad opmal hars Tabl I. I Cas 1, h ra of a dras h MRL 1 s for h MWM ad hars ar almos h sam rgardlss of h valu of. I ohr words, h MRL 1 for h MWM har s approxmaly h sam as ha of h har wh s a osa. For xampl, wh =., h MRL 1 for Cas 1 of h MWM har s 14 whl ha of h har s 15. Wh =.5 owards, h MRL 1 s of boh hars for Cas 1 ar xaly h sam. mlarly, Cas, alhough dffrs bw h MRL 1 s of h MWM ad hars xs, h dffrs ar small ad sgfa. Th MRL 1 rds of h wo hars Tabl I, mod abov ar also obsrvd Tabls II o IV. Thrfor, w olud ha h opmal MWM ad opmal orol hars hav smlar prforma dg ou-oforol sgals. I s worh og ha s h MRL 1 valus for Cass 1 ad of h MWM ad hars ar abou h sam, for h sam valu of, a graphal llusrao of MRL 1 vrsus, basd o h rsuls Tabls 1 4 s lss magful as h plos for h MWM ad hars wll ovrlap. Thr s glgbl dffr h MRL 1 s of h wo hars wh has h sam valu. VII. COCLUIO Basd o h MRL omparso, w olud ha boh h opmal MWM ad opmal hars ar qually ff dg shfs. Th MRL prformas of h opmal MWM ad opmal hars ar omparabl dg ou-of-orol sgals for Cas I (a shf o varabl) ad Cas (a shf wo varabls). Th MRL a b usd as a alrav or as a sodary rro o h RL, h valuao of h prformas of MWM ad hars. I hs sudy, w oly osdr h us of MRL o valua h prformas of opmal MWM ad opmal hars, whh volv wo qualy hararss (p = ). Furhr rsarh a b mad by rasg h umbr of qualy hararss, p sudyg h MRL prformas of h wo hars. I s also usful o sudy h xsg mhods ad suggs supror os, drmg whh of h p dffr varabls orbu o a ou-oforol sgal, wh o ours. CKOWLDGMT Ths work s fudd by h Uvrs as Malaysa, hor Trm Rsarh Gra, o. 34/PMTH/63936 ad h UM Fllowshp. RFRC [1] D. C. Mogomry, Iroduo o asal Qualy Corol, 4h d., w York: Joh Wly, 1. [] H. Hollg, Mulvara Qualy Corol, I C. shar, M. W. Hasay, ad W.. Walls, ds., Thqus of asal alyss, w York: MGraw-Hll, [3] C. W. Champ ad.. Rgdo, omparso of h Markov ha ad h gral quao approahs for valuag h ru lgh dsrbuo of qualy orol hars, Commu. as - mul. Compua, vol. (1), pp , [4] M. H. L ad M. B. C. Khoo, Opmal sasal dsg of a mulvara WM har basd o RL ad MRL, Commu. as - mul. Compua., vol. 35, pp , 6a. [5] M. H. L ad M. B. C. Khoo, Opmal sasal dsg of a mulvara CUUM har basd o RL ad MRL, I. J. Rlab. Qual. af. g., vol. 13, pp , 6b. [6] L. Y. Ch ad C. K. Low, omparso of h RL prformas of opmal MWM ad opmal orol hars, Fal yar proj, hool of Mahmaal s, Uvrs as Malaysa, Malaysa, ug. 8. [7] C.. Lowry, W. H. Woodall, C. W. Champ ad.. Rgdo, mulvara xpoally wghd movg avrag orol har, Thomrs, vol. 34, pp ,

4 TBL I MRL 1 OF TH OPTIML MWM D OPTIML CHRT WITH MRL = 37, P =, ρ = D HIFT, =.1 Cas 1 (a shf o varabl) MWM (r =.8, δ H = 5.6) (k =.9, H = 18.5) Cas (a shf wo varabls) MWM (r =.8, H = 5.6) (k =.9, H = 18.5) δ TBL II MRL 1 OF TH OPTIML MWM D OPTIML CHRT WITH MRL = 37, P =, ρ = D HIFT, =.3 Cas 1 (a shf o varabl) MWM (r =.15, δ H = 7.1) (k =.5, H = 11.64) Cas (a shf wo varabls) MWM (r =.15, H = 7.1) (k =.5, H = 11.64) δ

5 TBL III MRL 1 OF TH OPTIML MWM D OPTIML CHRT WITH MRL = 37, P =, ρ = D HIFT, =.5 Cas 1 (a shf o varabl) MWM (r =.6, δ H = 1.6) (k =.35, H = 8.68) Cas (a shf wo varabls) MWM (r =.6, H = 1.6) (k =.35, H = 8.68) δ TBL IV MRL 1 OF TH OPTIML MWM D OPTIML CHRT WITH MRL = 37, P =, ρ = D HIFT, = 1. Cas 1 (a shf o varabl) MWM (r =.16, δ H = 11.53) (k =.675, H = 5.16) Cas (a shf wo varabls) MWM (r=.16, H=11.53) (k=.675, H=5.16) δ [8].. Prabhu ad G. C. Rugr, Dsgg a mulvara WM orol har, J Qual Thol, vol. 9, 8-15, [9] R. B. Crosr, Mulvara gralzaos of umulav sum qualy orol shms, Thomrs, vol. 3, pp , [1] J. D Haly, o o mulvara CUUM produrs, Thomrs, vol. 9, pp , [11] H. ga ad J. Zhag, Mulvara umulav sum orol hars basd o projo pursu, a. a, vol. 11, pp , 1. [1] J. J. Pgallo, ad G. C. Rugr. Comparsos of Mulvara CUUM Chars, J Qual. Thool, vol., pp , 199. [13] P. Qu ad D. M. Hawks, oparamr mulvara umulav sum produr for dg shfs all dros, J. Roy. as. o. r. D, vol. 5, pp , 3. [14] W. Woodall ad M. M. ub, Mulvara CUUM qualy orol produrs, Thomrs, vol. 7, pp. 85-9,