Performance of cumulative count of conforming chart with variable sampling intervals in the presence of inspection errors

Transcription

1 Joual of Idustial ad Systems Egieeig Vol. 10, special issue o Quality Cotol ad Reliability, pp Wite (Febuay) 2017 Pefomace of cumulative cout of cofomig chat with vaiable samplig itevals i the pesece of ispectio eos Mohammad Sabe Fallah Nezhad 1*, Yousof Shamstaba 1 1 Idustial Egieeig Depatmet, Yazd Uivesity, Ia Fallahezhad@yazd.ac.i, y.shamstaba@gmail.com Abstact I high quality idustial pocesses, the cotol chat is desig based o cumulative cout of cofomig (CCC) items is vey useful. I this pape, the pefomace of CCC- chat with vaiable samplig itevals (CCC- VSI chat) i the pesece of ispectio eos is ivestigated. The efficiecy of CCC- VSI chat is compaed with CCC- chat with fixed samplig iteval (CCC- FSI chat). The compaiso esults show thatthe VSI scheme ca pefoms bette tha the FSI scheme. I additio, aalysis ad discussio of the esults ae peseted to illustate the effect of iput paametes o the pefomace of CCC- VSI chat. Keywods: high quality pocesses; CCC- chat; vaiable samplig iteval; ispectios eos; aveage time to sigal 1- Itoductio As oe of the basic Statistical Pocess Cotol (SPC) tools, cotol chat is useful i maitaiig stability ad impovig quality though vaiability eductio i the poductio pocess. The cumulative cout of cofomig (CCC) cotol chat is based o the cumulative umbe of cofomig items betwee two cosecutive ocofomig items that is the adom vaiable with the geometic distibutio (Calvi, 1983; Goh, 1987).This cotol chat is mostly applied fo high- quality pocesses. I the automated ad discete maufactuig systems, vey low level of o-cofomig items is poduced. As a esult the CCC chat has eceived cosideable attetio fom the idusty (Joekes et al., 2016).The CCC chat is developed based o the CCC chat which cosides the cumulative cout of cofomig items util obsevig a fixed umbe '''' of ocofomig oes that follows the egative biomial distibutio(ohta et al.,2001; Kudo et al.,2004). The scheme of vaiable samplig iteval (VSI) is desiged i ode to impove the efficiecy of the CCC cotol chat (Liuet al., 2006).The cuet eseaches o the applicatio of VSI scheme have deoted the bette pefomace of this scheme i compaiso with FSI scheme. X cotol chat with vaiable samplig itevals was poposed by some eseaches (see fo example, Reyolds et al.,1988; Reyolds ad Aold, 1989; Ruge ad Pigatiello J, 1991; Ruge ad Motgomey, 1993; Ami ad Mille, 1993; Zhag et al.,2012). Apaisi ad Hao, (2001) ad Villalobos et al. (2005) studied a VSI multivaiate shewhat chat. Reyolds et al., (1990) ad Luo ad Wag (2009) ivestigated the VSI- CUSUM cotol chat. VSI- EWMA chats have bee studied by some eseaches (see fo example, Shamma et al., 1991; Saccucci et al, 1992;Castagliolaet al, 2006; Castagliolaet et al, 2006). *Coespodig autho. ISSN: , Copyight c 2017 JISE. All ights eseved 78

2 Some studies have bee pefomed fo VSI- CCC (CCC VSI ) chat ad they cocluded that thei poposed desig is moe efficiet tha FSI- CCC (CCC FSI chat). Fist, Liu et al. (2006) poposed CCC chat with vaiable samplig itevals (CCC VSI chat). Che et al. (2011) ivestigated CCC chat with vaiable samplig itevals ad vaiable cotol limits (CCC VSI/VCL chat) ad Che (2013) studied CCC chat with vaiable samplig itevals fo coelated samples (GCCC VSI chat). Zhag et al. (2014) ivestigated the pefomace of CCC VSI chat with estimated cotol limits. The stategy of usig 100% ispectio is poposed fo the implemetatio of the CCC chat ad may studies o this chats have assumed that ispectio is pefect accuate, but ispectio eos exist i the pocess. Buke et al.(1995) discussed ispectio eos ad thei impact o cotol chats ad deoted thei impotat effect o the esults. Lu et al. (2000) calculated the adjusted cotol limits fo the CCC chat i the pesece of ispectio eos based o the elatioship betwee the tue ad obseved values of the ocofomig popotio. Raja et al. (2003) desiged a pocedue to set cotol limits fo CCC chats cosideig the ispectio eos to obtai the maximum ARL. Some othe studies about ispectio eos have bee doe by Case (1980), Lidsay (1985), Suich (1988),Huag et al. (1989), Suich (1990), Johso et al.(1991), Cheg ad Chug(1994), Wag ad Che (1997, Rya (2011), Nezhad ad Nasab (2012), Fallahezhad ad Babadi(2015) ad Fallah Nezhad et al.(2015). The goal of this pape is to develop a model to coside the ispectio eos i implemetatio of vaiable samplig iteval scheme fo CCC- cotol chat. So, this pape cosides the adjusted cotol limits fo the CCC-VSI chat that ca educe fo the effects of ispectio eos. I sectio 2, the CCC- VSI Chat i the pesece of ispectio eoshas bee descibed.i sectio 3, the compaiso study betwee the CCC- VSI chat ad the CCC- FSI chat is pefomed ad the esults ae elaboated usig sesitivity aalysis. I sectio 4, a pactical case study fo the implemetatio of the CCC- VSI chat i the pesece of ispectio eos is descibed. Fially, we have cocluded the pape. 2- Desciptio of the CCC- VSI chat 2-1- Notatios p 0t p 1t p 0 p1 the obseved value of the i-cotol ocofomig factio, the obseved value of the out-of-cotol ocofomig factio, the tue value of the i-cotol ocofomig factio, the tue value of the out-of-cotol ocofomig factio, α the pobability of false alam, disied X i d j d IL i the cumulative cout of items ispected afte obsevig the (i-1) th ocofomig item util the i th ocofomig item (icludig the last ocofomig item), the umbe of diffeet itevals fo the CCC- VSI chat, j =1,2,,. Samplig iteval legths fo the CCC- VSI chat,, i.e., the time betwee ispectios of two cosecutive items (d <d -1 < <d 2 <d 1 ) the samplig iteval legth fo the CCC- FSI chat, i =1,2,,-1. the iteval limits i the CCC- VSI chat which divide the egio betwee UCL ad 79

3 LCL ito sub-egios 1; 2;... ; I I I ( IL IL IL ) < < <, L i the samplig iteval legth which is used to obtai X i, ARL a ATS a ATS av ATS af the adjusted aveage u legth, the adjusted aveage time to sigal, the adjusted i-cotol ATS of the CCC- VSI chat, the adjusted i-cotol ATS of the CCC- FSI chat, ' ATS the adjusted out-of-cotol ATS of the CCC- VSI chat, av ATS af I q j q j the adjusted out-of-cotol ATS of the CCC- FSI chat, impovemet facto, defied as I = ATS / ATS ' V the pobability that poit X i falls egio I j whe the pocess ocofomig factio is p 0, the pobability that poit X i falls withi the egio I j whe the pocess ocofomig factio is p CCC- VSI cotol chat i the pesece of the ispectio eos The elatioship betwee the tue ad obseved value of ocofomig popotio i pesece of ispectio eos is as follows (Buke et al., 1995): t ( 1 ) ( 1 ) p = p e + p e 2 t 1 ' F, (1) The obseved (estimated) o-cofomig factio is p t ad p is the tue value of ocofomig popotio, while ( ) ad ( ) deote, espectively, the pobability of classifyig a cofomig item as ocofomig ad the pobability of classifyig a ocofomig item as cofomig. So, we have: i cotol state: p0 = p0t ( 1 e2 ) + ( 1 p0t ) e1 out of cotol state: = ( 1 ) + ( 1 ) p p e p e 1 1t 2 1t 1 If the acceptable isk of false alam isα desied, the the cotol limits ad the cete lie ca be detemied as follows (Xie et al., 2012): UCL 1 i= i 1 (1 0 ) i p t p0t = 1 αdisied / 2 1 (2) 80

4 i 1 (1 ) 0.5 CL i p0t p0t = i= 1 (3) LCL i= i 1 (1 0 ) i p t p0t = αdesied / 2 1 Lu et al. (2000) poposed the adjusted acceptable isk of false alam i the pesece of ispectio eos. Thus a cotol chat is modified to povide a fist type eo that is close to the oe ude eo-fee ispectio i ode to educe the impact of ispectio eos. I the pesece of ispectio eos, ca be obtaied as followig, = (5) As the esult, i the pesece of ispectio eos, the adjusted cotol limits ad the cete lie ca be detemied as follows: (4) UCL a 1 i= CL a i= LCL a i= i 1 (1 0 ) i p p0 1 = α disied / 2 1 i 1 (1 0 ) i p p0 = i 1 (1 0 ) i p p0 = α desied / 2 1 (6) (7) (8) The ARL (aveage u legth) is defied as the aveage umbe of poits plotted util eceivig a outof-cotol sigal. Thus, ARL ca be obtaied as followig, ARL a 1 = UCLa i 1 1 p (1 p) LCL 1 a a i ANI (aveage umbe of items) is defied as the expected value of the umbe of items ispected util the chat sigals a alam.ani a ca be computed fo CCCG- FSI ad CCCG- VSI chat by applyig the followig equatio: ANI a = ARLa (10) p Whe the CCC- VSI chat is applied, the the time betwee ispectios of two cosecutive items would be d, d,..., d (d > d >... >d ). These iteval legths should be detemied cosideig the (9)

5 pactical coditios of poductio system. As a example, the miimum value of iteval legth is ot less tha the time lag betwee poductios of two successive items. The maximum value of iteval legth ca be obtaied with egads to the maximum amout of time that is allowed fo the pocess to u without ispectio. The iteval limits IL 1,IL 2,...,IL ae detemied i the CCC- VSI chat, so that the iteval betwee UCL a ad LCL a is divided ito diffeet itevals I 1, I 2,..., I. Thus followig famewok is used fo samplig fom the pocess, d1,x i 1 I1 = ( IL1, UCLa ) d2,x i 1 I2 = ( IL2, IL1 a). Li =.. d,x i 1 I = (L CLa, IL ) IL, IL,..., The iteval limits 1 2 IL 1ca be detemied as the followig that F -1 is ivese fuctio of the egative biomial distibutio fuctio with ad p 0 paametes : =( +1 < )=( +1) ( ) & = # $ 1 1 % (1 ) '- # $ 1 1 % (1 ) '- '()* +,./*- & '(./* = # $ 1 1 % (1 ) '- =1 0( ) 2 '()* +, =0 - (1 ) 2 So we have: =0 - (1 ) 2 = (11) 4- =0 - (1 4 ) 2 (12) 82

6 This scheme cotiues util the IL values falls betwee UCL a ad LCL a as followig: LCL < IL < IL <... < IL < IL < UCL a a (13) ATS is the aveage legth of time that is eeded to obseve a sigal i a cotol chat. Also, ATS af ad ATSaV ca be detemied as followig, ATSaF = ANIa d = ARLa d (14) p ATS av d1q1 + d2q dq = ARLa p q + q q 1 2 (15) 3- Pefomace compaisos betwee the CCC- VSI ad the CCC- FSI chat The pefomace of CCC- VSI is compaed with the CCC- FSI chat i this sectio. Note that the same values of ocofomig factio p 0 ad false alam pobabilityα ae assumed fo both the CCC FSI ad the CCC VSI chat. I ode to compae these chats, the desig paametes fo the CCC- VSI ad the CCC FSI chat ae detemied so that the equatio ATS af = ATS av is satisfied at the i cotol state. O the p ( > p ) othe had, whe the pocess ocofomig factio chages to 1 0, the values of ' ATS av should be evaluated. The cotol chat with smalle value of out-of-cotol the bette pefomace. Let, thus, ATS af = ATS av ATS ad ' a ' af ATS will have d d q + d q d q d q + d q d q = = q + q q 1 α (16) It is assumed that the samplig iteval legth of the CCC- FSI chat is adjusted to be equal 1 ( d = 1), the values of ( d1, d2,..., d ) ad ( q1, q2,..., q ) ae detemied so that Eq. (16) is satisfied the the matched CCC- VSI ad CCC- FSI chat ae obtaied that have the same i-cotol value of ATS. The, whe the ocofomig factio chages to p 1, the pefomace of the CCC- VSI chat ca be aalyzed by computig the value of I, that is equal to the atio of out-of-cotol ATS a of the CCC- VSI ad the CCC FSI chat: I ATS d q + d q d q = = ATS q q q ' ' ' ' av ' ' ' ' af (17) Based o Equatio (17), if the value of Impovemet facto is less tha 1.00, vaiable samplig iteval scheme ca poduce a sigal moe quickly tha fixed samplig iteval scheme whe the pocess is out of cotol. So, whe I is less tha 1.00, it deotes that the CCC- VSI chat pefoms bette tha the CCC- FSI chat. The values of ' q j ca be calculated as followig, 83

7 UCLa 1 x 1 q 1 = p1 (1 p1) x= IL x 1 q p (1 p ) 1 = x= IL x IL x IL 2 x 1 q 1 = p1 (1 p1) x IL = + IL 1 x 1 q = p1 (1 p1) x= LCL 1 1 a + x x (18) The pefomace of CCC- VSI chat is aalyzed based o the umbe of samplig iteval () assumig equal pobabilities fo each iteval: 1 α q1 = q2 =... = q = (19) By substitutig d=1 i Equatio (16), we have, 1 α 1 α = d1q1 + d2q dq = ( d1 + d d) d1 + d d = (20) 4- Compaative study of CCC- VSI chat i the pesece of ispectio eos I this pape, we apply the iput data i the umeical study of Liu et.al (2006)fo compaiso study. This data is as followig: α disied = , p0t = ad samplig iteval legths (d 1, d2,..., d ) with the fixed value of d = 1ca be chose as follows: = 2, d = 1.9, d = 0.1; = 3, d = 1.9, d = 1, d = 0.1; = 4, d = 1.9, d = 1.2, d = 0.8, d = 0.1; = 5, d = 1.9, d = 1.5, d = 1, d = 0.5, d = 0.1; Impovemet factos fo diffeet pocess shifts Now, we study the pefomace of CCC- VSI chat i the pesece of ispectio eos fo diffeet value of pocess shifts ad seveal values of e 1 ad e 2.Fist, the value of coespodig impovemet 84

8 factos I ae computed ad the esults ae show i Table 1. As ca be see whe the ocofomig atio (p 1t /p 0t ) iceases the, the impovemet facto I deceases, ad thus the CCC- VSI chat pefoms bette tha CCC- FSI chat i the pesece of the ispectio eos. Table 1. Impovemet factos fo diffeet pocess shifts with =2 ad =3 p 1t /p 0t e 1 e Whe e 1 (the pobability of classifyig a ocofomig item as cofomig) iceaseas, the impovemet facto I also iceseas ad with iceasig e 2, the value of impovemet facto iceases. Thus it is cocluded that the supeioity of CCC- VSI chat ove CCC- FSI chat impoves by iceasig the espectio eos Impovemet factos fo diffeet CCC- VSI cotol chat I this subsectio, we fix the umbe of samplig itevals (=2), ad pocess shift (p 1t /p 0t =2) the fo diffeet possible values of paamete, the esults ae show i Table 2. The impovemet facto deceases by iceasig the paamete i the all cases. 85

9 Table 2. Impovemet factos fo diffeet CCC- VSI cotol chat with =2 ad (p 1t /p 0t =2) e 1 e Impovemet factos fo diffeet umbes of samplig itevals I ode to ivestigate the oveall pefomace of CCC- VSI chat based o the umbe of samplig itevals, we fix the paamete =3, ad pocess shifts (p 1t /p 0t ) = 2. The esults i Table 3 idicate that fo diffeet values of e 1 ad e 2, the umbe of samplig itevals is efficiet o the impovemet facto, I. fo example, if e 1 = ad e 2 =0.0001, the CCC- VSI chat with =2, is moe efficiet ad if e 1 =0.005 ad e 2 =0.0001, the CCC- VSI chat with =5, is moe efficiet. 86

10 Table 3. Impovemet factos fo diffeet umbes of samplig itevals with =3 ad p 1t /p 0t =2 e 1 e Impovemet factos based o diffeet legths of samplig iteval Based o the above aalysis, we ivestigate the effect of iteval legth o the pefomace of CCC- VSI chat whe the umbe of samplig itevals is =2.Fou diffeet cases of samplig iteval legths ae aalyzed. As ca be see i Table 4 the lage values fo the diffeeces betwee iteval legths, (d 1, d 2 ) leads to bette pefomace of CCC- VSI chat. Also, i all cases, the value of I is less tha 1, thus the pefomace of CCC- VSI chat is bette tha CCC- FSI chat i all cases. 87

11 Table 4. Impovemet factos based o diffeet legths of samplig iteval with =2,=3 ad p 1t /p 0t =2 (d 1,d 2 ) d 1 -d 2 e 1 e (1.9,.1) (1.7,0.3) (1.5,0.5) (1.2,0.8) (1,1)=FSI Impovemet factos fo diffeet pobability allocatios The above oveall pefomace of CCC- VSI chat is aalyzed based o the equal i cotol pobability allocatios. I ode to ivestigate the oveall pefomace of CCC- VSI chat whe the coditio q 1 = q 2 = = q is ot satisfied, we fix =2 ad d 1 =1.9, ad oly chage the values of i cotol pobability, q 1 as poposed by Liu et al. (2006). It should be oted that q 1 +q 2 =1-α. The value of d 2 ca be obtaied usig the followig equatio: 1 α d (21) 1q1 d2 = > 0 q 2 As show i Table 5, whe (q 1 -q 2 ) deceases, impovemet facto 1 deceases ad the pefomace of CCC- VSI chat i the pesece of the ispectio eos becomes bette. 88

12 Table 5. Impovemet factos fo diffeet pobability allocatios with =2, =3 ad p 1t /p 0t =2 q q 1 -q 2 e 1 e (0.1,0.8973) (0.2,0.7973) (0.3,0.6973) (0.4,0.5973) ( , ) Coclusio I maufactuig techology, may poductio pocesses today ae poducig a vey small popotio of ocofomig items. Thus, may pocess cotol methods have bee poposed, such as CCC chat that has eceived cosideable attetio fom the idusty. I this pape, we have ivestigated the pefomace of CCC- VSI cotol chat i the pesece of the ispectio eos fo high quality pocesses. Some sesitivity aalysis was doe ad the esults demostated that the CCC- VSI chat is moe efficiet tha CCC- FSI chat ad whe the paamete iceases the, the efficiecy of CCC- VSI chat will be ehaced. Also the supeioity of CCC- VSI iceases by iceasig the diffeece betwee the iteval legths ad uifom pobability allocatio is moe efficiet. Fo futue woks, we suggested developig the CCC- chat with the vaiable samplig itevals ude goup ispectio i the pesece of ispectio eos. 89

13 Refeeces Ami, R. W., & Mille, R. W. (1993). A obustess study of X chats with vaiable samplig itevals. Joual of Quality Techology, 25, Apaisi, F., & Hao, C. L. (2001). Hotellig's T2 cotol chat with vaiable samplig itevals. Iteatioal Joual of Poductio Reseach, 39(14), Buke, R. J., Davis, R. D., Kamisky, F. C., & Robets, A. E. (1995). The effect of ispecto eos o the tue factio o-cofomig: a idustial expeimet. Quality Egieeig, 7(3), Calvi, T. (1983). Quality Cotol Techiques fo" Zeo Defects". IEEE Tasactios o Compoets, Hybids, ad Maufactuig Techology, 6(3), Case, K. E. (1980). The p cotol chat ude ispectio eo. Joual of Quality Techology, 12(1), 1-9. Castagliola, P., Celao, G., & Fichea, S. (2006). Evaluatio of the statistical pefomace of a vaiable samplig iteval R EWMA cotol chat. Quality Techology & Quatitative Maagemet, 3(3), Castagliola, P., Celao, G., Fichea, S., & Giuffida, F. (2006). A vaiable samplig iteval S2-EWMA cotol chat fo moitoig the pocess vaiace. Iteatioal Joual of Techology Maagemet, 37(1-2), Che, S.-L., & Chug, K.-J. (1994). Ispectio eo effects o ecoomic selectio of taget value fo a poductio pocess. Euopea Joual of Opeatioal Reseach, 79(2), Che, Y.-K. (2013). Cumulative cofomace cout chats with vaiable samplig itevals fo coelated samples. Computes & Idustial Egieeig, 64(1), Che, Y. K., Che, C. Y., & Chiou, K. C. (2011). Cumulative cofomace cout chat with vaiable samplig itevals ad cotol limits. Applied stochastic models i busiess ad idusty, 27(4), Fallahezhad, M., & Babadi, A. Y. (2015). A New Acceptace Samplig Pla Usig Bayesia appoach i the pesece of ispectio eos. Tasactios of the Istitute of Measuemet ad Cotol, 37(9), Fallah Nezhad, M. S., Yousefi Babadi, A., Owlia, M. S., & Mostafaeipou, A. (2015). A ecusive Appoach fo lot Setecig poblem i the Pesece of ispectio eos. Commuicatios i Statistics- Simulatio ad Computatio, (just-accepted), Goh, T. (1987). A cotol chat fo vey high yield pocesses. Quality Assuace, 13(1), Huag, Q., Johso, N. L., & Kotz, S. (1989). Modified Dofma-Steett sceeig (goup testig) pocedues ad the effects of faulty ispectio. Commuicatios i Statistics-Theoy ad Methods, 18(4), Joekes, S., Smeka, M., & Righetti, A. F. (2016). A compaative study of two poposed CCC- chats fo high quality pocesses ad thei applicatio to a ijectio moldig pocess. Quality Egieeig,

14 Johso, N. L., Kotz, S., & Wu, X.-Z. (1991). Ispectio eos fo attibutes i quality cotol (Vol. 44): CRC Pess. Kudo, K., Ohta, H., & Kusukawa, E. (2004). Ecoomic Desig of A Dyamic CCC Chat fo High- Yield Pocesses. Ecoomic Quality Cotol, 19(1), Lidsay, B. G. (1985). Eos i ispectio: itege paamete maximum likelihood i a fiite populatio. Joual of the Ameica Statistical Associatio, 80(392), Liu, J., Xie, M., Goh, T., Liu, Q., & Yag, Z. (2006). Cumulative cout of cofomig chat with vaiable samplig itevals. Iteatioal Joual of Poductio Ecoomics, 101(2), Lu, X., Xie, M., & Goh, T. (2000). A ivestigatio of the effects of ispectio eos o the u-legth cotol chats. Commuicatios i Statistics-simulatio ad Computatio, 29(1), Luo, Y., Li, Z., & Wag, Z. (2009). Adaptive CUSUM cotol chat with vaiable samplig itevals. Computatioal Statistics & Data Aalysis, 53(7), Nezhad, M. F., & Nasab, H. H. (2012). A ew Bayesia acceptace samplig pla cosideig ispectio eos. Scietia Iaica, 19(6), Ohta, H., Kusukawa, E., & Rahim, A. (2001). A CCC chat fo high yield pocesses. Quality ad Reliability Egieeig Iteatioal, 17(6), Raja, P., Xie, M., & Goh, T. (2003). Optimal cotol limits fo CCC chats i the pesece of ispectio eos. Quality ad Reliability Egieeig Iteatioal, 19(2), Reyolds J, M. R., & Aold, J. C. (1989). Optimal oe-sided Shewhat cotol chats with vaiable samplig itevals. Sequetial Aalysis, 8(1), Reyolds, M. R., Ami, R. W., & Aold, J. C. (1990). CUSUM chats with vaiable samplig itevals. Techometics, 32(4), Reyolds, M. R., Ami, R. W., Aold, J. C., & Nachlas, J. A. (1988). Chats with vaiable samplig itevals. Techometics, 30(2), Ruge, G.C., & Motgomey,D. (1993). Adaptive samplig ehacemets fo Shewhat cotol chats. IIE tasactios, 25(3), Ruge, G. C., & Pigatiello J, J. J. (1991). Adaptive samplig fo pocess cotol. Joual of Quality Techology, 23(2), Rya, T. P. (2011). Statistical methods fo quality impovemet: Joh Wiley & Sos. Saccucci, M. S., Ami, R. W., & Lucas, J. M. (1992). Expoetially weighted movig aveage cotol schemes with vaiable samplig itevals. Commuicatios i Statistics-simulatio ad Computatio, 21(3), Shamma, S. E., Ami, R. W., & Shamma, A. K. (1991). A double expoetially weigiited movig aveage cotol pocedue with vaiable samplig itevals. Commuicatios i Statistics-simulatio ad Computatio, 20(2-3),

15 Suich, R. (1988). The c cotol chat ude ispectio eo. Joual of Quality Techology, 20(4), Suich, R. (1990). The effects of ispectio eos o acceptace samplig fo ocofomities. Joual of Quality Techology, 22(4), Villalobos, J. R., Muñoz, L., & Gutieez, M. A. (2005). Usig fixed ad adaptive multivaiate SPC chats fo olie SMD assembly moitoig. Iteatioal Joual of Poductio Ecoomics, 95(1), Wag, R.-C., & Che, C.-H. (1997). Miimum aveage factio ispected fo cotiuous samplig pla CSP-1 ude ispectio eo. Joual of Applied Statistics, 24(5), Xie, M., Goh, T. N., & Kualmai, V. (2012). Statistical models ad cotol chats fo high-quality pocesses: Spige Sciece & Busiess Media. Zhag, M., Nie, G., & He, Z. (2014). Pefomace of cumulative cout of cofomig chat of vaiable samplig itevals with estimated cotol limits. Iteatioal Joual of Poductio Ecoomics, 150, Zhag, Y., Castagliola, P., Wu, Z., & Khoo, M. B. (2012). The vaiable samplig iteval X chat with estimated paametes. Quality ad Reliability Egieeig Iteatioal, 28(1),