DESIGNING OF GENERALIZED TWO-PLAN SYSTEM WITH REFERENCE SAMPLING PLAN. Department of Statistics, Bharathiar University, Coimbatore,Tamilnadu,India.

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1 DESIGNING OF GENERALIZED TWO-PLAN SYSTEM WITH REFERENCE SAMPLING PLAN 1 K. K. Sush nd K. Vnth Xv 2 1 Pofsso nd Hd of th Dptmnt Dptmnt of Sttstcs, Bhth Unvsty, Combto,Tmlndu,Ind. 2 Rsch Schol, Dptmnt of Sttstcs, Bhth Unvsty, Combto,Tmlndu,Ind. ABSTRACT: Ths pp poposs gnlzd two-pln smplng systm wth multpl pttv goup smplng pln s th fnc pln, dsgntd s GTPMRGSS (n; cn, ct), s ntoducd. Th ffcncy of GTPMRGSS (n; cn, ct) wth spct to smll smpl szs hv bn stblshd ov th ttbuts schm. Th smplng nspcton schm wll b usful whn tstng s costly nd dstuctv. Th dvntgs of th smplng nspcton schm ov ttbuts sngl, doubl nd vous fnc plns dscussd. Tbls constuctd consdng vous combntons of ccptbl nd lmtng qulty lvls nd wth th optng tos. KEYWORDS: Accptbl Qulty Lvl, Lmtng Qulty Lvl, Mkov modl, Optng to, Two-Pln smplng systm. INTRODUCTION A smplng systm conssts of two o mo smplng plns nd uls fo swtchng btwn thm to chv th dvntgous ftus of ch. Govndju nd Submn (1992) hv poposd two pln systm whch s n ccptnc smplng pln nvolvng only noml nd tghtnd nspcton. A noml nspcton s cd out whn th good qulty poducts nd t s swtchd to tghtnd nspcton whn th s dtoton n qulty of poducts. Th two pln systm usng dffnt swtchng ct to chv th dsd dscmnton on optng chctstc (OC) cuv ws l nvstgtd by Dodg (1965), Hld nd Thygod (1966) nd Stphns nd Lson (1967). Clvn n (1977) hs poposd th zo ccptnc numb tghtnd-noml-tghtnd whch s spcl cs of th two pln systm towds pplcton of ttbuts chctstcs. Vjyghvn nd Soundjn n (1996) hs nvstgtd th pfomnc of noth typ of two-pln systm dsgntd s TNT (n; c1, c2) schm. Th dvntg of th two pln systms s tht t gvs dsd potcton wth mnmum smpl sz. Th smplng systm utlzs two zo ccptnc numb sngl smplng plns of dffnt smpl szs, togth wth swtchng uls to buld up th should of th optng chctstc (OC) cuv. Assumng ccptnc numb, c, to tk vlus oth thn zo, th gnlzd two pln smplng systm cn b dsgntd s GTPS (n, kn; c), whch fs to GTPS schm wh th noml nd tghtnd sngl smplng plns hv th ccptnc numb, c, but, on tghtnd nspcton, th smpl sz s n(< kn). Anoth wy of dfnng th Two Pln smplng systm s to sy tht th noml nd tghtnd plns utlz th sm 24

2 smpl sz n but wth dffnt ccptnc numbs. Ths typ of gnlzd two pln systm cn b dsgntd s GTPS (n; cn,ct) whch fs to GTPS wh th noml nd tghtnd plns hv fxd smpl sz n but dffnt ccptnc numbs, sy cn nd ct (< cn) spctvly. In ths pp, Gnlzd Two Pln Systm wth Multpl Rpttv Goup Smplng pln s th fnc pln, dsgntd s GTPMRGSS (n; cn,ct) s ntoducd, wh n s th smpl sz und th fnc pln, nd cn nd ct th ccptnc constnts cospondng to noml nd tghtnd plns spctvly. Th ffcncy of GTPMRGSS (n; cn,ct) wth spct to smll smpl szs hs bn stblshd ov th ttbut smplng schm. Th poposd pln my mo conomcl nd cost ffctv on whch s usd fo dstuctv tms to fnd th non-dfctv tm usng th vous smplng pln. Swtchng uls bsd plns gnlly dvntgous thn clsscl smplng pln n tms of smpl sz ffcncy nd should ffct on OC cuv. GENERALIZED TWO PLAN SYSTEM Dodg (1969) hs poposd smplng nspcton nvolvng noml nd tghtnd nspcton plns whch usully fd s gnlzd two-pln systm. Ths systm s lgly ncopotd n MIL-STD-105E (1989), whch foms n ntgtd smplng nspcton systm guntng th consum tht th outgong qulty wll b btt thn th spcfd AQL nd t th sm tm ssung th poduc tht th sk fo jcton wll b smll fo poducts of AQL qulty o btt. Kulmn (1992) hs dsgnd two-pln swtchng systm nvolvng ccptbl nd lmtng qulty lvls. Th pocdu wth p of plns gvs n ovll OC cuv tht gnlly ls n btwn th OC cuv of th noml nd tghtnd plns n Two-Pln swtchng systm. Blmul nd Ch-Hyuck Jun (2009) hv md contbutons to dsgnng of vbls two-pln systm by mnmzng th vg smpl numb (ASN). Sush (1993) hs poposd pocdus to slct ctn fnc plns ndxd though poduc nd consum qulty lvls consdng flt nd ncntv ffcts. Th concpt of Rpttv Goup Smplng (RGS) pln ws ntoducd by Shmn (1965) whch ccptnc o jcton of lot s bsd on th ptd smpl sult of th sm lot. Th opton of th pln s sml to tht of squntl smplng pln. Accodng to Shmn, th RGS pln gvs mnmum smpl sz s wll s dsd potcton. Rmsmy (1983) md contbutons towds th constucton of RGS plns. Gu Shnk nd Josph (1993) hv dvlopd noth nw RGS pln s xtnson of th Condtonl RGS pln n whch th ccptnc o jcton of lot on th bss of ptd smpl sults s dpndnt on th outcom of nspcton und RGS nspcton systm of th pcdng lots. In ths pp, th poposd pln wll b dsgntd s Multpl Rpttv Goup Smplng pln. Slcton of Smplng Pln: Condtons fo Applcton 1. Th poducton s stdy nd th sults on cunt, pcdng nd succdng lots bodly ndctv of contnung pocss. 2. Lots submttd substntlly n th od of poducton. 3. Th poduct coms fom souc n whch th consum hs confdnc. 25

3 Optng Pocdu Swtchng uls fo gnlzd Two-pln Systms : Noml to Tghtnd Whn noml nspcton s n ffct, tghtnd nspcton shll b nsttutd (whn s out of m conscutv lots o btchs jctd on ognl nspcton (s<m)). Tghtnd to Noml Whn tghtnd nspcton s n ffct, noml nspcton shll b nsttd (whn d conscutv lots o btchs ccptd on ognl nspcton). A dgmmtc psntton fo th swtchng uls to gnlzd two-pln systm s shown n Fgu1. s out of m conscutv lots jctd stt Noml Inspcton Tghtnd Inspcton d conscutv lots ccptd Totl of m lots jctd on tghtnd Dscontnu nspcton Fgu 1. Swtchng uls fo Gnlzd Two-Pln Systm A numb of mpotnt msus of pfomnc to b dtmnd nd usd n th vluton of OC functon whch wll b dscussd. = th popoton of lots xpctd to b ccptd und noml nspcton. PN PT IN IT = th popoton of lots xpctd to b ccptd und tghtnd nspcton. = th xpctd popoton of lots nspctd on noml nspcton. = th xpctd popoton of lots nspctd on tghtnd nspcton. Usng bov msus, th compost optng chctstc functon cn b dtmnd s, P (p) = IN PN + IT PT (1) Wh 26

4 nn = th (vg) smpl sz fo th noml nspcton pln. nt = th (vg) smpl sz fo th tghtnd nspcton pln. Th mthod fo dvng vous msus of pfomnc fo th Gnlzd Two- Pln Systm lso studd. MARKOV CHAIN AND SAMPLING SYSTEM In od to psnt th gnlzd two-pln systm s Mkov chn, st of stts (vnts) s dfnd so s to compltly dscb th opton of th systm. Ths vnts mutully xclusv snc t ny tl, th stt of th systm s dscbd by on nd only on vnt. Moov, ths vnts hv Mkov popty n th sns tht t ny tl th pobblty of bng n ptcul stt dpnds only on th stt occupd t th pvous tl. Th vnts nd dfntons gvn blow: N = th vnt tht th noml nspcton s n ffct (=1, 2,, m) T = th vnt tht th tghtnd nspcton s n ffct (=1, 2,, d) PN = th pobblty of th systm bng n stt N (=1,2,, m) PT = th pobblty of th systm bng n stt T (=1,2,, d) Fo th sk of convnnc, lt us dnot PN nd PT s nd b spctvly fo vlutng th bov msus. Th tnston pobblts of th gnlzd two-pln systm shown n Fgu 2. 27

5 1-1- N1 N2 Ns-1 Ns Nm b T1 b T2 b b b T3 T4 Td 1-b 1-b 1-b 1-b b Fgu 2. Tnston Pobblts fo Gnlzd Two-Pln Systm Und th ssumpton of constnt poduct qulty p, t s not dffcult to s tht th Mkov chn dfnd by th tnston mtx n Tbl 1 s ducbl nd podc, nd thus suffcnt condtons stsfd fo th Mkov chn to possss non-zo sttony dstbuton. Thus th Mkov chn s godc,., th lmtng dstbuton s th sm s th sttony dstbuton. Ths condtons dscussd fo th dvton of th followng ncssy fomuls. It cn b sn tht noml nspcton s dvd by th unon of th vnts, N, = 1,2,..,m. Thus th xpctd popoton of tm tht noml nspcton s n ffct s gvn s, IN = m P N 1 (2) Smlly, th xpctd popoton of tm tht tghtnd nspcton s n ffct s gvn s, IT = d P T 1 (3) All th pobblts n ths xpssons lmtng stt pobblts. As dscussd bov, du to th godcty of th Mkov chn, th lmtng pobblts sm s th sttony of wll known qutons, Psj = 1 P P S ; j = 1,2,. j 28

6 Stt -1 Intntonl Jounl of Mthmtcs nd Sttstcs Studs j 1 nd P 1 (4) Sj wh Pj th on stp tnston pobblts fom stts S to Sj. Ps s sttony nd hnc lmtng pobblty of th systm bng n stt S. DERIVATION OF THE OC AND ASN FUNCTIONS Th OC nd ASN functons cn b dvd usng th tnston pobblts gvn n Tbl 1, nd qutons (4) yld qutons (5) though (12), Stt Tbl 1. Tnston pobblty mtx fo Gnlzd Two pln systm PN = (1-) -1 PN1 ; = 1,2,3,,s (5) = -s (1-) s-1 PN1; = s+1,s+2,,m PT = b -1 PT1 ; = 1,2,,d (6) All pobblts cn now b vlutd usng th condton tht th sum of ll pobblts quls to on, on cn gt,, IN + IT =1 (7) IN = s 1 ( 1 ) 1 P N1 m s 1 s (1 ) s 1 P N1 on smplfcton, IN = I T = s 2 m s 2 PN 1[1 (1 ) (2 1)] P N1 (1 )(1 b d (1 b) b m s 1 d )(1 ) s 1 (8) (9) Substtutng qutons (8) nd (9) n (7), w hv PN1 = (1 b) b A B wh A = (1-b)b d [1+(1-) s-2 (2- m-s+2-1)] B = (1- m-s+1 )(1-b d )(1-) s-1 Agn substtutng quton (10) n (8) nd (9), w hv d (10) 29

7 N1 N2 N3 Ns Ns+1 Ns+2 Nm T1 T2 T3 Td-1 Td N1 1- N2 1- Ns-1 1- Ns 1- Ns+1 1- Nm-1 1- Nm 1- T1 1-b b T2 1-b b Td-1 1-b b Td 1-b IN = (11) 30

8 IT = (12) wh, 1 (1 ) (1 s 2 (2 )(1 ) m s 2 d 1 b (1 b) b d m s 2 Substtutng qutons (11) nd (12) n (1) wth s PN nd b s PT, th compost OC functon obtnd s s 1 1) P (p) = P N P T (13) Wh, PN = Pobblty of ccptnc und th noml nspcton. PN = p(d cn /n,p) PT = Pobblty of ccptnc und th tghtnd nspcton. PT = p(d ct /n,p) Not tht wh μ nd τ th vg numb of lots nspctd usng noml nspcton bfo gong to tghtnd nspcton nd vg numb of lots nspctd usng tghtnd nspcton bfo gong to noml nspcton spctvly. Slcton of Gtpmgss Th MRGS pln s n xtnson of Condtonl Rpttv Goup Smplng pln n whch ccptnc o jcton of lot on th bss of ptd smpl sults s dpndnt on th outcom of nspcton und Rpttv Goup Smplng nspcton systm of th pcdng lots. Futh thy dvd th fomul fo OC nd ASN functons. An ttmpt hs bn md to modl nd nlys th dynmcs of th poposd nspcton systm though GERT ppoch. Optng Pocdu Followng th nottons sml to thos of Shmn, th Multpl Rpttv Goup Smplng pln s cd out though th followng stps; Stp 1: Dw ndom smpl of sz n nd dtmn th numb of dfctvs d found th n. Stp 2: Accpt th lot, f d c1. Rjct th lot, f d>c2. Stp 3: If c1<d c2, pt th stps 1, 2 nd 3 povdd succssv pvous lots 31

9 ccptd, und RGS nspcton systm, othws jct th lot. Both th plns chctzd though fou pmts, nmly n, c1, c2 nd ccptnc ct. H, t my b notd whn c1=c2 th sultng pln s sngl smplng pln. Also whn =0, ths pln bcoms RGS pln du to Shmn. Futh, t my b notd tht th condtons fo th pplctons of th poposd pln s sm s Shmn RGS pln. OPERATING CHARACTERISTICS FUNCTION Th optng chctstc functon P(p) of Multpl Rpttv Goup Smplng pln s dvd by posson modl s, P(p) = p ( 1 p ) (1 p ) c c p c p wh, c1 p = p[d c1] = 0! c2 pc = p[c1<d<c2] = 0! c1-0! nd x = np. Dsgnton GTPMRGSS (n; c1n, c2n; c1t,c2t) nd fs to Gnlzd Two Pln Systm of typ I (n;cn,ct) wh th noml MRGS pln hs smpl sz n nd ccptnc numb c1n, c2n (c1n<c2n) nd th tghtnd MRGS pln hs smpl sz n nd ccptnc numb c1t, c2t(c1t<c2t, c1t c1n nd c2t c2n). Dsgnng Gtpmgss wth Dffnt Pmts Th OC cuv fo GTPMRGSS cn b constuctd usng tbl 2. Ths cn b don by dvdng ch nty fo th gvn vlus of c1n, c2n, c1t, c2t nd by th cospondng vlu of smpl sz n. Th sult of ch dvson s th numb of non-confomts p unt fo whch th P(p) s shown blow. Fo xmpl, whn n=25, c1n=1, c2n=4, c1t=1, c2t=2, whn =1 dvson of ch of th nts n th c1n=1, c2n=4, c1t=1, c2t=2, whn =1 ow of tbl 2 by 25 lds to th followng vlus, P(p) P Fo plottng OC cuv th Gnlzd Two Pln Multpl Rpttv Goup Smplng Systm (25; 1,4; 1,2) whn =1. 3) Th OC cuv hs bn obtnd usng SAS pogm fo th vlus gntd (Fgu. 32

10 Fgu 3. OC cuvs of Gnlzd Two Pln Multpl Rpttv Goup Smplng Systm Dsgnng Systms fo gvn p1, α, p2 nd β Tbl 3 cn b usd to dsgn Gnlzd Two Pln Multpl Rpttv Goup Smplng Systm (GTPMRGSS), whn two ponts on th OC cuv (p1, 1-α) nd (p2,β) gvn. To dsgn GTMRGSS clcult th Optng Rto (OR) = p2/p1. Fom tbl 3 on cn dtmn th vlu of OR whch s nst to th dsd to. Cospondng to th slctd OR vlus of c1n, c2n, c1t, c2t, nd np1 whn =1. Th smpl sz s dtmnd thus dvdng np1 by p1. Fo xmpl, lt p1=0.06, α=0.05, p2=0.16, nd β=0.05, clcult th Optng Rto (OR) = p2/p1 = 0.16/0.06 = Fom th tbl 3 th vlu of OR fo α= 0.05, β=0.05 whch s nst to th dsd to s Cospondng to ths slctd OR vlus s c1n=1, c2n=2, c1t=0, c2t=1, =1 nd np1= Th smpl sz s obtnd s n= np1/p1=1.131/0.06= Th dsd systm s GTMRGSS (19; 1,2 ; 0,1) whn =1. Constucton of Tbls Th xpsson fo pobblty of ccptnc of Gnlzd Two Pln Multpl Rpttv Goup Smplng Systm (GTMRGSS), und th ssumpton of Posson modl, th compost OC functon quton (13) s gvn s, PN= P (p) = c P p(1 pc) (1 p ) p p c N P T (14) 33

11 wh c1 N p = 0! c2 N pc = 0! c1 N - 0! nd x = np. PT= p(1 pc) (1 p ) p p c c (15) wh c1t p = 0! c2t pc = 0! c1t - 0! nd x = np. Fo vous ssumd vlus of c1n, c2n, c1t, c2t, s, m, d, nd P(p) th quton (13) s solvd wth quton (14) nd (15) fo np usng tton tchnqus fo dffnt vlus of c1n, c2n, c1t, c2t, s, m, d nd. Fom tbl 2 vous ncomng qulty lvls, outgong qulty lvls nd optng to vlus clcultd fo dffnt α nd β vlus whch gvn n tbl 3. CONCLUSION An ttmpt s md towds th concpt of Two Pln Multpl Rpttv Goup Smplng Systm (TPMRGSS) n whch dsposl of lot s on th bss of noml nd tghtnd plns. Posson unty vlus hv bn tbultd fo wd ng of pln pmts. Whnv on fnds th OC cuv fo ttbut pln to b unstsfctoy, thn ts shp cn b mpovd by usng two-pln systm povdd h, whch llows swtchng btwn two knds of smplng systm sml to th cs of ttbuts noml nd tghtnd nspcton schms. Th dsgn pmts such s numb of ccptnc numbs dtmnd by stsfyng th poduc, consum, nd ngns t vous ncomng nd outgong qulty lvls. On compng th pfomnc of th smplng pocdus th tchnqus povd to b btt thn xstng ons. Futh studs my consd nvolvng vbl smplng pln nd lblty of smplng plns. Acknowldgmnts Th uthos thnkful to th unknown f fo hs commnts towds vson of ths pp. Th fst utho s thnkful to Unvsty Gnts Commsson, Nw Dlh fo povdng Unvsty Gnd Commsson On Tm Gnd (UGC-OTG). Th scond utho s thnkful to Bhth Unvsty uthots fo povdng ncssy fclts. REFERENCES Blmul,S. nd Ch-Hyuck Jun. (2009) Dsgnng of vbls two-pln systm by mnmzng th vg smpl numb, Jounl of Appld Sttstcs, 36, Clvn,T.W. (1977) TNT Zo Accptnc Numb Smplng, Amcn Socty fo Qulty Contol, Annul Tchncl Confnc Tnscton Phldlph Pnnsylvn Dodg, H.F. (1965) Evluton of Smplng Inspcton Systm hvng uls fo Swtchng btwn Noml nd Tghtnd Inspcton, Tchncl Rpot No.14, Th Sttstcs Cnt, Rutgs Stt Unvsty Nw Bunswck, Nw Jsy. Dodg, H.F. (1969) Nots on th Evoluton of Accptnc Smplng Plns-Pt I, Jounl of Qulty Tchnology, Vol.1 No.2: Gu Snk. nd Josph,S. (1993) GERT Anlyss of Multpl Rpttv Goup Smplng Plns, IAPQR Tnsctons, Vol.19 No.2:

12 Govndju,K. nd Submn.K. (1992) Slcton of tghtnd noml tghtnd systm fo gvn vlus of th ccptbl qulty lvl nd lmtng qulty lvl, Jounl of Appld Sttstcs, Vol. 19 Issu 2: Hld,A. nd Thygod,P. (1966) Th Compost Optng Chctstc Und Noml nd Tghtnd Smplng Inspcton by Attbuts, Bulltn of th Intntonl Sttstcl Insttut, Vol.41: Kulmn,V. (1992) Studs on Dsgnng Mnmum Inspcton Attbut Accptnc Smplng Plns, Ph.D. Thss, Bhth Unvsty, Tml Ndu, Ind. MIL -STD- 105E. (1989) Smplng Pocdus nd Tbls fo Inspcton by Attbuts, US Govnmnt Pntng Offc, Wshngton, DC. Rmsmy,M.M. (1983) Rpttv Goup Smplng (RGS) Pln, M.Phl. Thss, Bhth Unvsty, Combto, Ind. Shmn, R.E. (1965) Dsgn nd Evluton of Rpttv Goup Smplng Plns, Tchnomtcs, Vol.7 No.1: Sush, K.K. (1993) A study on Accptnc Smplng Pln usng Accptbl nd Lmtng Qulty Lvls, Ph.D.Thss, Bhth Unvsty, Tmlndu, Ind. Stphns, K.S. nd Lson, K. E. (1967) An Evluton of th MIL-STD 105D Systm of Smplng Plns, Industl Qulty Contol, Vol.23 No.7: Vjyghvn, R. nd Soundjn,V. (1996) Pocdus nd tbls fo th slcton of tghtnd noml tghtnd TNT (n; c1, c2) smplng schms, Jounl of Appld Sttstcs, Vol.23:

13 Pobblty of Accptnc s m d cn ct c1n c2n c1t c2t Tbl 2. Unty vlus fo Gnlzd Two Pln Systm (n;cn,ct) wth MRGS pln whn =1 Tbl 3. Optng Rto vlus fo Two Pln Systm (n; cn, ct) wth MRGS Pln whn =1 36

14 p2/p1 fo α =0.01 p2/p1 fo α =0.05 α = 0.05 α = 0.05 α = 0.05 α = 0.01 α = 0.01 α = 0.01 s m d cn Publshd ct c1n by Euopn c2n c1t Cnt c2t fo β Rsch = 0.10 Tnng β = 0.05 nd Dvlopmnt β = 0.01 UK β = ( β = 0.05 β =