GREY PREDICTIVE PROCESS CONTROL CHARTS

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1 The 4th Internatonal Conference on Qualty Relablty Augut 9-th, 2005 Bejng, Chna GREY PREDICTIVE PROCESS CONTROL CHARTS RENKUAN GUO, TIM DUNNE Department of Stattcal Scence, Unverty of Cape Town, Prvate Bag, Rhode Gft, Rondeboh 7707, Cape Town, South Afrca It notced modern manufacturng envronment where the lfe cycle of product ha decreaed rapdly cutomzed hort-run manufacturng proce become qute common for achevng cutom atfacton (Pan, Any manufacturng proce nevtable poee the nherent rom varablty n proce telf accordngly the product meaurement varaton therefore treated a a tochatc proce, where partal parameter can not oberved. In clacal tattcal control theory, t aumed that for a table producton proce f the product meaurement were wthn control the ntrnc varablty poe no mpact to product qualty. Stattcal etmaton nference on proce parameter are typcally large-ample baed therefore the evaluaton of proce tablty of requre long-run. Such character of tattcal control cheme obvouly not utable to today hort-run bune envronment. The root caue the large-ample requrement baed on tard tattcal etmaton nference theory. In th paper, we wll nvetgate a mall-ample orented grey predctve control cheme whch developed n term of grey dfferental equaton theory (Fu, 992 Wen, Th new control approach doe not mpoe the tablty retrant to the producton proce eay to be mplemented for contant montorng product qualty. A an llutratve example, grey predctve Shewhart chart CUSUM control chart are dcued. Fnally, we propoe a grey fuzzy predctve control cheme. Keyword: Stattcal qualty control, grey fuzzy predctve control, grey Shewhart control chart, grey CUSUM control chart. Introducton Stattcal proce control methodology ha enjoyed t ucceful ndutral applcaton n the pat eghty year, poneered by A. Shewhart. Although the world ha experenced three ndutralzaton qualty control theory methodology ha evolved ubtantally the tattcal proce control alo evolved never lot t crtcal role n modern qualty control. In the proce perperctve, t aumed that all the varaton n a ytem are from one of the caue thee are the natural varaton or not natural varaton. The caue of natural or rom varaton are called common caue. Th due to the condton preent n the proce wll occur all the tme unle the proce changed. It hould be noted that reducng varaton due to thee caue normally cotly. The caue for not natural or rom varaton are called pecal caue or agnable caue. Thee caue are not alway preent. They generally occur when omethng doe not go rght. Thee caue can uually be elmnated relatvely ealy. When they are elmnated, only natural or rom varaton reman. When the proce operatng wth only natural

2 2 varaton preent the proce ad to be n a tate of tattcal control called table proce. In toady peedng up globalzaton, jut a Pan ponted out (2002, the lfe cycle of product ha decreaed rapdly cutomzed hort-run manufacturng proce become qute common for achevng cutom atfacton. Therefore, the table manufacturng envronment, typcally referrng to mave quantty producton, may not alway ext for a compettve bune. The term table mplctly mpled the large-ample tattcal theory correctly reflected the mave producton ndutral realty. In other word, the ucce of tattcal proce control etablhed t foundaton on large-ample baed tattcal etmaton nference theory. Wthout dtrbutonal aumpton, partcularly, aymptotc normalty of tattc ued, there were not any upport. Facng the challenge of today untable bune envronment, t neceary to walk out from the umbrella of large-ample baed tattcal theory by changng our atttude lookng at the real world from a dfferent angle. Becaue ytem dynamc can be treated from the vewpont of the degree of nformaton avalablty, t poble for u to walk out from the umbrella of large ample tattc develop a mallample baed predctve proce control theory methodology baed on the grey dfferental equaton theory, partcularly, GM(, model. The tructure of the paper followng: Secton 2 ued to revew GM(, Model; ecton 3 to nvetgate a grey predctve Shewhart control chart ecton 4 contrbutng to the grey predctve CUSUM control Chart; n Secton 5 a grey fuzzy predctve control cheme propoed the fnal ecton gve a hort concluon. 2 A Revew of GM(, Model It well-know fact that dfferental equaton play the key role n modern control theory. People often have a percepton that dfferental equaton can only be appled to the contnuou dfferentable ytem. The etablhment of grey dfferental equaton utlzng mall-ample dcrete data equence propoed by Deng (985 dramatcally rehaped the way of mathematcal-tattcal thnkng. Therefore, t crtcal to revew the GM(, model. Theorem 2.. Let dcrete data equence ( 0 ( 0 Z { = Z (, R,,2,, N} + = L ( a dummy varable whch repreent the ndex varable atfyng the followng grey dfferental equaton repectvely, where ( 0 β + α Z + Z =, = 2, L, N ( ( ( ( 0 0 Z ( + = Z + + Z, = 2, L, N (2 2 Accordngly, the frt-tage leat-quare parameter etmator are

3 The 4th Internatonal Conference on Qualty Relablty Augut 9-th, 2005 Bejng, Chna ( α ˆ β T T ˆ, = X X X y (3 N where ( 0 ( 0 ( Z Z X= 2 M M + 2 ( 0 ( 0 ( Z Z 3 2 ( 0 ( 0 ( Z N Z ( N (4 x N Z Z 2 Z ( Z = 3 2 M ( Z ( Z N N N N Thu the frt-tage repone equaton can be rewrtten a ( 0 exp ( ˆ α γ β Then we enter the econd-tage leat-quare etmaton Z = + (6 ( ˆ α ˆ γ (5 T T T, = D D D z (7 N where ( ˆ β ( ˆ β2 exp exp D= x M M exp ˆ ( βn (8

4 z 4 Z Z ( 0 ( ( 0 ( x 2 N = Z ( 0 ( Therefore after two-tage leat-quare fttng, the etmated repone functon M N ( 0 ˆ ˆ exp ( ˆ α γ β (9 Z = + (0 It notced that the two-tage etmaton amng at obtan a repone functon on (0 the frt-order grey dervatve level,.e., ( Z ( = dz ( d the reaon why the parameter β wa kept unchanged after the frt-tage etmaton becaue the exponental form at Z ( ( level. 3 A Grey Predctve Shewhart Control Chart Shewhart control chart are the prmal mot ntutve tattcal proce control tool. A typcal chart can be drawn a Fg.. Upper control lmt Sample qaulty charactertc Center lne Lower control lmt Sample Number or tme Fg. A General Shewhart Control Chart. If our prmary concern of the qualty manufacturng to mantan the qualty charactertc jut wthn control, wth the central locaton denoted by c, the lower control lmt denoted by l the upper control lmt denoted by u. Furthermore the (0 ( 0 amplng obervaton of the proce denoted by X = { X (, =,2,, n}. Denote ( the tranformed data equence a ( x (0 = { x 0 (, x 0 ( = X (0 ( c, =,2,, n}. (0 ( 0 Decompoe the equence x = { x ( } nto the potve component negatve component a the followng.

5 The 4th Internatonal Conference on Qualty Relablty Augut 9-th, 2005 Bejng, Chna (0 (0 (0 + x f 0 < x < u c x ( = ( 0 otherwe x ( (0 (0 (0 x f l c < x < 0 = (2 0 otherwe Denote the upper under-control equence by (0 + (0 + X = { x ( k > 0, k {, 2,, n} } denote the lower under-control equence by (0 (0 X = { x ( jk > 0, jk {, 2,, n} }. Accordngly, two predctve curve can be (0 etablhed baed on X + (0 X repectvely, denoted them by + + x ( + k x ( = a k b a ( ( exp ( ( exp ( x k x (0 j a k b a b b + = + a a (3 (4 repectvely. Then the decon rule for the grey predctve Shewhart control chart wll ( + ( + be that the proce under-control f only f x ( n+ x max( k < u c or ( ( x ( n+ x jmax( k < c l. It alo poble to ue GM(, model to perform the out-control predcton f the ampled data equence contan more than 4 out-control data. Denote the out-control ( 0 > ( 0 > ( 0 > ( 0 > equence by X = { x ( q(, x ( q( 2,, x ( q( K, K 4} (0 correpondngly, the tme equence of out-control wll be Q = { q(, q( 2,, q( K }. Then the out-control tme wll be ( b (0 q b qˆ ( k+ = q ( exp( aqk + aq a ( 0 ( ( qˆ ( k+ = qˆ ( k+ qˆ ( k q q (5 It a well-known theoretcal foundaton that ARL ued for etablhng the optmal degn of Shewhart control chart. However, a a matter of fact, n mall batch proceng, ARL not meanngful. Therefore, n term of Equaton to decrbe the potentally next out-of control tme pont take preventve acton.

6 4 A Grey Predctve CUSUM Control Chart 6 Page (962 propoed the cumulatve um (CUSUM control chart a an alternatve to the Shewhart X-bar control chart. It preferred by many uer of the CUSUM to tardze the varable x before performng the calculaton. Th method of conderng the problem of controllng a normally dtrbuted tardzed tattc wa propoed by Luca Croer [?]. The tardzed value of x y x µ σ 0 = (6 where µ 0 the proce target value, σ the (aumed known tard devaton, th x the obervaton for the CUSUM cheme. The th obervaton ( x mght repreent a ngle readng or the average of a number of obervaton from a degnated routne amplng plan. A CUSUM cheme cumulate devaton more than k (tardzed unt from the target mean value. Thu k erve a the reference value of the cheme. Th then lead to the tardzed Two-Sded CUSUM defned a: + C max + 0, y k C = + C = max 0, y k+ C where the frt for detectng potve mean hft the econd for detectng negatve mean hft. Stardzng the CUSUM ha two man advantage. Frt many CUSUM chart can now have the ame value of k h, the choce of k h are not dependent on σ. The other advantage that the tardzed CUSUM lead to a CUSUM for controllng varablty. It obvou that the grey predctve CUSUM control chart can be developed n a very mlar manner a that n Shewhart Control chart becaue of the well-etablhment of the two-ded (tardzed CUSUM control chart. Denote the upper-ded undercontrol equence a Cu = { cu ( q( = C ( 0 ( 0 + ( < h, =,2,, I > 3 q } the lower-ded ( 0 ( 0 under-control equence a Cl = { cl ( p( j = C < h, j=,2,, J > 3 p j }. Denote the (0 upper lower under-control equence by Q = { q(, q( 2,, q( I } (0 P = { p(, p( 2,, p( J } repectvely. Then n term of mlar way a Equaton (5, two under-control tme predctve equaton can be etablhed. It then a routne to check t the proce under-control va a CUSUM chart. 5 A Grey-Fuzzy Predctve Control Chart The qualty of conformance how well the product conform to the pecfcaton tolerance requred by degn (Montgomery, 986. Qualty proce n a manufacturng envronment n nature to ecure the atfactorly proceng a product n term of degn pecfcaton tolerance level. (7

7 The 4th Internatonal Conference on Qualty Relablty Augut 9-th, 2005 Bejng, Chna Qualty-conformance a vague concept therefore can be repreented by a fuzzy event, hgh conformance to pecfcaton, denoted by Q. Ceng (994 dcued the etablhment of fuzzy control chart n term of α -level cut et concept. However, we wll propoe a predctve control cheme baed on memberhp grade of the proce ample. Let a degn pecfcaton lmt of a product be [ L, U ]. Then accordng to the pecfcaton-tolerance requrement, a memberhp functon of hgh-conformance Q gven by U -L U -L lx f x 6 2 (8 U - L 5( U -L µ ( x r( x f x Q = otherwe where 0 lx a monotone-ncreang functon 0 rx a monotonedecreang functon wth U L l r U L = = 2 2. It notced that the meaurement x here the hfted one,.e., x = X L where X the actual obervaton value. Let a proceed tem have the followng meaurement equence by equal tme (0 (0 amplng X = { X (, =,2,, n}, where n 4. Then accordng to the memberhp functon µ Q of hgh-conformance, the meaurement ample can be converted nto memberhp grade equence µ ( (0 { ( (0 (, (0 ( (0 (,,2,, Q x = µ x x = X L = n Q }. Fnally, let u etablh the predctve equaton accordng to Theorem 2. va memberhp grade equence: ( ˆ µ x 0 ( = ˆ α + ˆ γ exp ˆ β, =,2,, n Q% L (9 Then accordng to Equaton (2, the predctve value of n + can be calculated, then the qualty (for example ( 0 perfect f 0.75 < ˆ µ ( ( Q% x n+ ( 0 ( under coptrol f 0.50 < ˆ µ (.75 Q% x n+ ( 0 quetonable f 0.25 ˆ ( < µ x ( n 0.50 Q% + ( 0 out of control f ˆ µ ( x ( n Q% ( 0 ( ˆ µ x n+ Q% at tme In ummary, we propoed th grey-fuzzy predctve control cheme baed on the followng three fact: ( qualty of conformance telf a fuzzy event therefore ung a memberhp to decrbe t behavor may be very approprate; (2 GM(, model poee hgh predctve power, accordng to the workng experence, the predctve (20

8 8 value wll not have relatve error larger than 5% a long a the predctve tme range le than n /3 therefore, one-tep predcton n general very afe to tell the qualty conformance tatu of the real proce; (3 GM(, modelng on memberhp grade equence µ ( (0 { ( (0 (, (0 ( (0 (,,2,, Q x = µ x x = X L = n Q } avod the hortage of data ncreae the model effcency becaue the grade value are all potve. 6 Concluon In th paper, we have developed two mall-ample baed grey predctve proce control chart: grey Shewhart control chart CUSUM control chart. Furthermore, we propoe a grey-fuzzy control chart whch n general more logcal more effcently utlzng the data nformaton becaue t doe not need to plt data equence nto upper (potve lower (negatve de. Furthermore, the memberhp functon can be characterzed by conderng both product pecfcaton tolerance parameter (proce capablty. A to the dentfcaton role of root caue affectng the proce out of control, the GM(,h model can be ued. We have ome fundamental dcuon n the paper (Guo, Reference. Y. T. Ceng. Fuzzy Qualty Control. The Publhng Houe of Gu Zhou Scence Technology, the People republc of Chna, R. Caulcutt. The Rght the Wrong of Control Chart, Appled Stattc, Vol. 44, No. 3, pp , J. L. Deng, Control Problem of Grey Sytem. Sytem Control Letter, Vol., No 6, March, E. R. DeVor, T. Chang J. W. Sutherl. Stattcal qualty Degn Control: Contemporary Concept Method, Macmllan, Inc, L. Fu. Grey Sytem Theory Applcaton, Bejng: The Publhng Houe of Scence Technology, 99 (n Chnee. 6. R. Guo. A Grey Modelng on Covarate Informaton n Reparable Sytem. To appear n the Proceedng of ICQR Conference Augut 2005, Bejng, J. M. Luca R. B. Croer. "Fat Intal Repone for CUSUM Qualty-Control Scheme: Gve Your CUSUM a Head Start," Technometrc, Vol. 24, pp , J. M. Luca M. S. Saccucc. Exponentally Weghted Movng Average Control Scheme: Properte Enhancement. Technometrc, Vol. 32. pp -2, D. C. Montgomery. Introducton to Stattcal Qualty Control, 3 rd edton, John Wley & Son, Inc, E. S. Page. A Modfed Control Chart wth Warnng Lne, Bometrka, Vol. 49, No. /2, 7 76, 962.

9 The 4th Internatonal Conference on Qualty Relablty Augut 9-th, 2005 Bejng, Chna. R. Pan. Stattcal Proce Adjutment Method for Qualty Control n Short-Run Manufacturng. PhD The, the Pennylvana State Unverty, Department of Indutral manufacturng Engneerng, S. B. Vardeman M. J. Jobe. Stattcal Qualty Aurance Method for Engneer, John Wley & Son, Inc, K. L. Wen. Grey Sytem: Modellng Predcton, Tawan: Yang' Scentfc Reearch Inttute, 2004.

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