CERTAIN RESULTS ON TIGHTENED-NORMAL-TIGHTENED REPETITIVE DEFERRED SAMPLING SCHEME (TNTRDSS) INDEXED THROUGH BASIC QUALITY LEVELS

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1 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Fb p-issn: CERTAIN RESULTS ON TIGHTENED-NORMAL-TIGHTENED REPETITIVE DEFERRED SAMPLING SCHEME (TNTRDSS) INDEXED THROUGH BASIC QUALITY LEVELS V.Kavyaasu 1 & A.Mankandan 2 1Assstant Pofsso, 2 Rsach Schola Dpatmnt of Statstcs, Bhaatha Unvsty, Combato , Tamlnadu, Inda. Cospondng addsss: 1kavyaas23@gmal.com, 2 mchal.amk10@gmal.com *** Abstact - Th psnt wok, amns Tghtnd- Nomal-Tghtnd Samplng Schm wth Rpttv Dfd Samplng plan as th fnc plan, dsgnatd as (TNTRDSS)-(n;, u2; v1, v2; ) ndd though poduc s sk and consum s sk fo spcfd Accptabl Qualty Lvl (AQL) and Lmtng Qualty Lvl (LQL) usng Posson dstbuton. Unty valus hav bn tabulatd to facltat th constucton and slcton of th schm. Fw llustatons w povdd fo th poposd wok. Kywods: Tghtnd-Nomal-Tghtnd, Rpttv Dfd Samplng, Opatng Chaactstc cuv, Accptabl Qualty Lvl (AQL) and Lmtng Qualty Lvl (LQL). 1. INTRODUCTION Accptanc samplng s an ffctv Qualty Contol tool, whch dals wth th pocdus n whch dcson to accpt o jct lots o pocss basd on th amnaton of sampls. Accptanc samplng s playng a ky ol n mpovng th qualty. Th basc am of any ndusts n th wold s to mpov th qualty of th poducts. Th hgh qualty of poducts may hav th hgh pobablty of accptanc. Accodngly samplng plans a wdly usd n manufactung ndusts, phamacutcal poducts and so on spcfcally n th aas of complanc and safty nspcton of th poducts. Accodng to Schllng (1982) th tghtndnomal-tghtnd (TNT) samplng schms a populaly usd n complanc samplng. fd sampl sz but wth two dffnt accptanc numbs. TNT samplng schm can also b accomplshd by mantanng th accptanc cta and swtchng btwn two sampl szs. Ths appoach s patculaly appalng wth zo accptanc numb plans. Calvn T.W (1977) has ntoducd Tghtnd Nomal Tghtnd Samplng Schm utlzng zo accptanc numb n two sngl samplng plans wth dffnt sampl szs, namly, n1 and n2 (< n1) togth wth th swtchng uls, whch s dsgnatd as TNT-(n1, n2; 0). Soundaaajan and Vjayaaghavan (1992) hav dvlopd TNT-(n1, n2; c) schm fo vaous nty paamts. Futh thy hav poposd anoth typ of TNT samplng schm usng on sampl sz and two accptanc numbs, dsgnatd as TNT-(n; c1, c2) schm. Ths schm utlzs th sngl samplng plan wth tghtnd plan of sampl sz n and accptanc numb c1 as wll as sngl samplng plan wth nomal plan of sampl sz n and accptanc numb c2. Radhakshnan and Svakumaan (2010) hav poposd TNT-(n1, n2; c) ndd though S Sgma Qualty Lvl-1 (SSQL-1) and S Sgma Qualty Lvl-2 (SSQL 2). Futh thy hav poposd TNT-(n1, n2; c) ndd though Poduc s Nano Qualty Lvl (PNQL) and Consum s Nano Qualty Lvl (CNQL). Subaman and Hadoss (2012) hav poposd Tghtnd-Nomal-Tghtnd systm fo gvn valus of th Accptabl Qualty Lvl and Lmtng Qualty Lvl usng Wghtd Posson dstbuton. 3. REPETITIVE DEFERRED SAMPLING PLAN 2. TIGHTENED NORMAL TIGHTENED SAMPLING SCHEME Tghtnd Nomal Tghtnd (TNT) s a samplng schm nvolvng swtchng btwn two samplng plans. MIL-STD-105D contans a samplng schm wth a In Rpttv Dfd Samplng plan, th accptanc o jcton of a lot n dfd stat s dpndnt on th nspcton sults of th pcdng o succdng lots und Rpttv Goup Samplng (RGS) nspcton. RGS s th patcula cas of RDS plan. 2016, IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 164

2 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Fb p-issn: Sanka and Mahopata (1991) has dvlopd th RDS plan whch s an tnson of th Multpl Dfd Samplng plan MDS-(c1, c2) whch was poposd by Rambt Vast (1980). Llly Chstna (1995) has gvn th pocdu fo th slcton of RDS plan wth gvn accptabl qualty lvls and also compad RDS plan wth RGS plan wth spct to opatng ato (OR) and ASN cuv. Sush and Samnathan, (2010) psnt a 4. Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS) H, th constucton and slcton of Tghtnd- Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS) fo dffnt paamt s and t a cad out und Posson modl. Th samplng plan was ndd though Incomng and Outgong Qualty lvls. Pfomanc masus a ndcatd fo Accptabl Qualty Lvl (AQL), Lmtng Qualty Lvl (LQL) and Indffnc Qualty Lvl (IQL). Ncssay tabls and pocdus w gvn fo dsgnng th schm though vaous nts of plan paamts Dsgnaton constucton and slcton of Rpttv Dfd Samplng (RDS) plan though Accptabl and Lmtng Qualty Lvls. Sush and Vntha Xav (2014) psnts a nw pocdu and tabls fo mnmum sum of sk of a Gnalzd Two Plan systm of typ GTPS- (n; cn, ct) wth Rpttv Dfd Samplng Plan as fnc plan ndd though Accptabl Qualty Lvl (AQL) and Lmtng Qualty Lvl (LQL). Stp 1: Daw a andom sampl of sz n and tst ach unt fo confomanc fo th spcfd qumnts Stp 2: Inspct und tghtnd nspcton usng th Rpttv Dfd Samplng plan wth sampl sz n and accptanc numb v1 and v2 (v2 > v1). If t lots n a ow a accptd und tghtnd nspcton, swtch to nomal nspcton (stp 3). Stp 3: Inspct und nomal nspcton usng th Rpttv Dfd Samplng plan wth sampl sz n and accptanc numb and u2 (u2 > ). Swtch to tghtnd nspcton, f an addtonal lots s jctd n th nt s. Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS)-(n;, u2; v1, v2; ) fs to a Tghtnd-Nomal-Tghtnd samplng nspcton wh th nomal RDS plan has a sampl sz n and accptanc numb, u2 ( < u2) and th tghtnd RDS plan has a sampl sz n and accptanc numb v1, v2 (v1 < v2, v1 and v2 u2) Opatng Pocdu fo TNTRDSS Th Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS) s cad out though th followng stps. N T N Thus, th Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS) has ght paamts. Thy a n;, u2; v1, v2, s, t and. Whn s = 4 and t = 5, th abov pocdu cosponds to MIL-STD-105D nvolvng only tghtnd and nomal nspcton. Whn =1 RDS plan ducs to Rpttv Goup samplng plan Opatng Chaactstcs Functon s t t s PT (1 PN )(1 PT )(1 PN ) PN PT (1 PT )(2 PN ) Pa ( p) s t t s (1 P )(1 P )(1 P ) P (1 P )(2 P ) Accodng to Calvn (1977) th psson fo OC functon of TNT s gvn by, T T N (1) Wh, PT and PN a pobablty of accptanc und tghtnd and nomal nspctons Dsgnng TNTRDSS fo gvn p 1, p 2, α and β Fo constucton and valuaton of th Tghtnd- Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS)-(n;, u2; v1, v2; ), th valus psntd n tabls w dvd und th pocdu statd by Duncan [1965]. Tabls a usd to dv ndvdual plan to mt spcfd valus of facton dfctvs and pobablty of accptanc. Tabls can b usd to slct Tghtnd-Nomal- Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS)-(n;, u2; v1, v2; ) fo gvn p1, p2, α and β by usng th followng stps. Stp 1: Spcfy p1 - AQL, p2 - LTPD, poduc sk (α) and consum sk (β). Stp 2: Comput th opatng ato s OR = p2 / p , IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 165

3 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Fb p-issn: Stp 3: Slct th valu p2 / p1 n Tabl 4 n th column fo appopat α and β that s naly qual to th computd OR. Stp 4: Dtmn th plan paamts of, u2, v1 and v2 cospondng to th valu of p2 / p1 locatd. Stp 5: Dtmn th valu of np1 fom Tabl 1 cospondng to th paamts, u2, v1 and v2. Stp 6: Th sampl sz of th schm s thn dtmnd by np1 by p1. Stp 7: Thus, th valus of n,, u2, v1 and v2 consttut th qud TNTRDSS-(n;, u2; v1, v2; ) schm. Th dsd systm s Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS)- (66; 3, 5; 2, 3) whn, s = 4, t = 5 and = Plottng th OC cuv of TNTRDSS Tabl 1 can b usd to obtan ght valus of p and Pa(p) to plot th OC Cuv of a gvn Tghtnd-Nomal- Tghtnd Rpttv Dfd Samplng Schm TNTRDSS-(n;, u2; v1, v2). Eampl 1: Suppos a Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS) s dsd wth Pa(p1) = 0.95 fo havng p1 = 0.01 and p2 = 0.75 thn, p1 = 0.02, α = 0.05, p2 = 0.08 and β = OR = p1 / p2 = 0.08 / 0.02 = Th opatng ato s OR = 4 n th tabl 1 whch s nast to th dsd ato s Cospondng plan paamts a = 3, u2 = 5, v1 = 2, v2 = 3 whn, s = 4, t = 5 and = 1 of computd OR. 4. Th np1 valu n Tabl 1 cospondng to th paamts, u2, v1 and v2 s Th sampl sz n = np1 / p1 = / 0.02 = Th OC cuv s obtand by dvdng th valus of np s gvn blow Pa(p). Pa(p) P Fgu 1: OC cuv fo TNTRDSS-(n;, u2; v1, v2) whn, = 1. Fgu 1 gvs th OC cuvs of Tghtnd-Nomal- Tghtnd Rpttv Dfd Samplng Schm TNTRDSS-(n;, u2; v1, v2; ) fo th paamt (66, 3, 5; 2, 3) and s = 4, t = 5 whn = 1. Fom th fgu, t s obsvd that th systm utlzs th nomal RDS plan whn th qualty of lot s good and th tghtnd RDS plan whn th qualty s poo. Futh, t s obsvd that th ncomng and outgong qualty of lot s good ath than th odnay RDS. 5. CONSTRUCTION OF TABLES Th psson fo pobablty of accptanc of Tghtnd-Nomal-Tghtnd Rpttv Dfd Samplng Schm (TNTRDSS)-(n;, u2; v1, v2; ), und th assumpton of Posson modl, th compost OC functon s gvn by quaton (1) wth 2016, IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 166

4 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Fb p-issn: P T v1 0 v2 v1! 0! 0! 0! 0! 0! (2) v v 2 1 0! v2 0! v1 v1 P N 0! u2 0! 0! u2 0! u2 0 0!! 0! 0! (3) Rspctvly wth < u2 and v1 < v2, Fo gvn valus of, u2, v1, v2, and Pa(p), th quaton 1 can b solvd aft substtutng th pssons PT and PN, to gt np usng th mthod of taton fo dffnt valus of s and t. Dffnt s and t, np valus fo gvn, u2, v1, v2, and Pa(p) a obtand and fom th np valus fo assumd valus of α and β, th OR = p2 / p1 a obtand and funshd n Tabl 1 to GLOSSARY OF SYMBOLS USED Pa (p) p p1 p2 p0 PT PN OR d 7. CONCLUSION = Pobablty of accptng a lot qualty p = Facton dfctv = Accptabl qualty lvl (AQL) = Lmtng qualty lvl (LQL) = Indffnc Qualty Lvl (IQL) = Popoton of lots pctd to b accptd usng Tghtnd nspcton. = Popoton of lots pctd to b accptd usng Nomal nspcton. = Opatng Rato = Dfctvs = Poduc s sk = Consum s sk Accptanc samplng s an ffctv Qualty Contol tool, whch dals wth th pocdus n whch dcson to accpt o jct lots o pocss basd on th amnaton of sampls. Th wok psntd n ths pap manly lats to th pocdu fo constucton and slcton of tabls fo Tghtnd-Nomal-Tghtnd plan wth Rpttv Dfd samplng plan as fnc plan whch s calld as TNTRDSS ndd though consum and poduc qualty lvls. Samplng schms dtmnd by ths mthod may hav btt sampl sz povdng by th qud nfomaton to accpt o jct th lot fo gvn qualty lvls (AQL, 1 α) and (LQL, β). Th Opatng Chaactstc (OC) cuv s also cad out fo th poposd samplng schm whch s dvd basd on act samplng dstbuton. Basd on OC cuv compason has mad fo nomal, tghtnd and TNTRDSS fo vaous s and t, to fnd that und s = 4 and t = 5 wll povd btt dscmnatng OC cuv. Th poposd TNTRDSS consds vaous basc qualty lvls, t s bnfcal fo th pacttons to us t n th qualty contol nvonmnts. It s advantagous to apply ths plan n ndusts fo ducng th nspcton cost and tm, patculaly n th aa of complanc tstng and spcally fo safty latd poduct. REFERENCES [1] Calvn, T. W., TNT Zo Accptanc Numb Samplng, Amcan Socty fo Qualty Contol Annual Tchncal Confnc Tansactons, Phladlpha, PA, pp , [2] Govndaaju, K., and Subaman, K., Slcton of a Tghtnd-Nomal-Tghtnd systm fo gvn valus of th accptabl qualty lvl and lmtng qualty lvl, Jounal of Appld Statstcs, Vol. 19 Issu 2, pp , [3] Radhakshnan, R., and Svakumaan, P. K., Constucton of Tghtnd-Nomal-Tghtnd schms ndd though S Sgma Qualty Lvls, Intnatonal Jounal of Advancd Opatons Managmnt, Vol. 2, Nos.1/2, pp , [4] G. Sanka, and B. N., Mahopata, GERT analyss of Rpttv Dfd Samplng plans, IAPQR Tansactons, 16, No. 2, pp , [5] Schllng, E. G., Accptanc Samplng n Qualty Contol, (Nw Yok, Macl Dkk), [6] Soundaaajan, V., and Vjayaaghavan, R., Constucton, slcton of Tghtnd-Nomal- Tghtnd samplng nspcton schm of typ TNT- (n1, n2; c), Jounal of Appld Statstcs, Vol. 19, pp , , IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 167

5 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Fb p-issn: [7] Soundaaajan, V., and Vjayaaghavan, R., Pocdus and tabls fo th slcton of Tghtnd-Nomal-Tghtnd TNT-(n; c1, c2) samplng schms, Jounal of Appld Statstcs, Vol. 23, pp , [8] Subman, K., and Hadoss, V., Slcton of a Tghtnd-Nomal-Tghtnd systm fo gvn valus of th Accptabl Qualty Lvl and Lmtng Qualty Lvl usng Wghtd Posson dstbuton, Intnatonal Jounal of Dcson Scncs, Rsk and Managmnt, 4(3/4), pp , [9] Sush, K. K., and Samnathan, R., Constucton and Slcton of Rpttv Dfd Samplng (RDS) plan though Accptabl and Lmtng Qualty Lvls, Intnatonal Jounal of Pu and Appld Mathmatcs, Vol. 65, No.3, pp , [10] Sush, K. K., and Vntha Xav, K., Dsgnng of Gnalzd Two Plan Systm wth Rpttv Dfd Samplng Plan as fnc plan usng mnmum sks, Intnatonal Jounal of Statstka and Mathmatka, Vol. 9, Issu 1, pp , , IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 168

6 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Nov p-issn: Tabl -1: Valu of np Tabulatd aganst s and t fo Gvn Pa (p) and Opatng ato Valu fo Gvn fo TNTRDSS-(n;, u2; v1, v2; ), Whn, = 1 s t u2 v1 v2 OR Pobablty of Accptanc p2 / p1 fo = 0.05 p2 / p1 fo = 0.01 =0.05 =0.05 =0.05 =0.01 =0.01 = =0.10 =0.05 =0.01 =0.10 =0.05 = , IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 169

7 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Nov p-issn: Tabl -2: Valu of np Tabulatd aganst s and t fo Gvn Pa (p) and Opatng ato Valu fo Gvn fo TNTRDSS-(n;, u2; v1, v2; ), Whn, = 2 s t u2 v1 v2 OR Pobablty of Accptanc p2 / p1 fo = 0.05 p2 / p1 fo = 0.01 =0.05 =0.05 =0.05 =0.01 =0.01 = =0.10 =0.05 =0.01 =0.10 =0.05 =

8 Intnatonal Rsach Jounal of Engnng and Tchnology (IRJET) -ISSN: Volum: 03 Issu: 02 Nov p-issn: Tabl -3: Valu of np Tabulatd aganst s and t fo Gvn Pa (p) and Opatng ato Valu fo Gvn fo TNTRDSS-(n;, u2; v1, v2; ), Whn, = 3 s t u2 v1 v2 OR Pobablty of Accptanc p2 / p1 fo = 0.05 p2 / p1 fo = 0.01 =0.05 =0.05 =0.05 =0.01 =0.01 = =0.10 =0.05 =0.01 =0.10 =0.05 = , IRJET Impact Facto valu: 4.45 ISO 9001:2008 Ctfd Jounal Pag 170