International Journal of Statistika and Mathematika, ISSN: E-ISSN: , Volume 9, Issue 1, 2014 pp 34-39

Transcription

1 Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9, Issue 1, Desgnng of Genealzed Two Plan System wth Reettve Defeed Samlng Plan as Refeence Plan Usng Mnmum Rsks K. K. Suesh 1, K. Vntha Xave 2 1 Pofesso and Head of the Deatment, 2 Ph.D Reseach Schola Deatment of Statstcs, Bhaatha Unvesty, Combatoe , Tamlnadu, INDIA. Coesondng Addesses: 1 sueshkk1@edffmal.com, 2 vntha.joby@gmal.com Reseach Atcle Abstact: Ths ae esents a new ocedue and tables fo mnmum sum of sk of a Genealzed Two Plan system of tye (n,c N,c T ) wth Reettve Defeed Samlng Plan as efeence lan ndexed though Accetable Qualty Level (AQL) and Lmtng Qualty Level (LQL). Tables ae constucted by consdeng vaous qualty levels, and llustatons ae also ovded fo eadymade selecton of lan aametes. Keywods: Accetable Qualty Level (AQL), Lmtng Qualty Level (LQL), Mnmum Rsks, Multle Defeed Samlng lan, Oeatng Chaactestc Cuve, Reettve Defeed Samlng Plan. 1. Intoducton Accetance samlng s a statstcal tool used to make decsons concenng whethe o not a lot of oducts should be eleased fo consume use. An accetance samlng lan s a statement egadng the equed samle sze fo oduct nsecton and the assocated accetance o ejecton ctea fo sentencng ndvdual lots. The ctea used fo measung the efomance of an accetance samlng lan, s usually based on the oeatng chaactestc (OC) cuve whch quantfes the sks fo oduces and consumes. The OC cuves lots the obablty of accetng the lot vesus the lot facton nonconfomng, whch dslays the dscmnatoy owe of the samlng lan. The basc accetance samlng lan called the sngle-samlng lan s wdely used n ndusty to nsect tems due to ts easness of mlementaton. A sngle samlng attbute nsecton lan calls fo accetance of a lot unde consdeaton. If the numbe of non-confomng unts found n a andom samle of sze n s less than o equal to the accetance numbe. Wheneve a samlng lan fo costly o destuctve testng s equed, t s common to foce the OC cuve to ass though a ont, say, (LQL, β). In ths ae Genealzed Two Plan System wth Reettve Defeed Samlng lan as efeence lan has been oosed. Dodge (1959) oosed a new samlng nsecton system namely twolan system. The two-lan system has a nomal as well as a tghtened lan whch has a tghte OC cuve comaed wth that of the nomal lan. A samlng system conssts of two o moe samlng lans and ules fo swtchng between them to acheve the advantageous featues of each. In geneal any samlng system of samlng nsecton nvolvng only nomal and tghtened nsectons wll be efeed to as a two-lan system. The tghtened nsecton can be used when the qualty of a oduct deteoated and nomal nsecton s used when the qualty s found to be good. Dodge(1965), Hald and Thyegod (1966) and Stehens and Lason(1967) have nvestgated the two-lan systems usng dffeent swtchng ctea to acheve the desed dscmnaton on the oeatng ato (OC) cuve and MIL-STD-105D (1963) systems usng the Makov chan aoach. Rombosk(1969) has nvestgated the oetes of a atcula tye such a two-lan system namely QS system whch was ognally oosed by Dodge(1967). Thus the genealzed two-lan system s vey useful to fnd efomance measues of a desed samlng system by substtutng numbes fo s, m, and d. Kualman(1992) has desgned two-lan swtchng system nvolvng accetable and lmtng qualty levels. The ocedue wth a a of lans gves an oveall OC cuve that geneally les n between the OC cuve of the nomal and tghtened lans n a Two-Plan swtchng system. Ths system s lagely ncooated n MIL-STD-105E (1989) whch foms an ntegated samlng nsecton system guaanteeng the consume that the outgong qualty wll be bette than the secfed AQL and at the same tme assung the oduce that the sk of ejecton wll be smalle fo oducts of AQL qualty o bette ones. The Reettve Defeed Samlng lan has been develoed by Shanka and Mohaata(1991) and ths lan s essentally an extenson of the Multle Defeed Samlng lan MDS-(c 1, c 2 ) due to Rambet Vaest(1981). In ths lan the accetance o ejecton of a lot n defeed state s Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9 Issue 1 Page 34

2 K. K. Suesh, K. Vntha Xave deendent on the nsecton esults of the ecedng o succeedng lots unde Reettve Gou Samlng (RGS) nsecton. So, RGS s a atcula case of RDS lan. Futhe Wotham and Bake (1976) have develoed Multle Defeed State Samlng (MDS) lans and also ovded tables fo constucton of lans. Suesh (1993) has oosed ocedues to select Multle Defeed State Plan of tye MDS and MDS-1 ndexed though oduce and consume qualty levels consdeng flte and ncentve effects. Llly Chstna (1995) has gven the ocedue fo the selecton of RDS lan wth gven accetable qualty levels and also comaed RDS lan wth RGS lan wth esect to oeatng ato(or) and ASN cuve. Suesh and Samnathan (2010) have studed the selecton of Reettve Defeed Samlng Plan though accetable and lmtng qualty levels. Golub(1953) has develoed a method of desgnng a sngle samlng lan when the samle sze s fxed and has gven an exesson fo c such that the sum of two sks namely oduce s sk ( ) and consume s sk (β) s mnmum. Mnmzng + β s same as maxmzng (1- ) + (1-β). The Golub s aoach fo sngle samlng lan has been extended by Soundaaajan(1981) unde osson model and hye geometc model. Soundaaajan(1978a,b)constucted the tables fo the selecton of Chs-1 lans unde osson model and also gven fo whch mnmzes the sum of oduce s and consume s sk fo secfed AQL and LQL when samle sze s fxed. Soundaaajan and Govndaaju (1982) have also studed the Chs-1 lan nvolvng mnmum sum of oduce s and consume s sk. Subaman(1991) has studed the selecton of sngle samlng lans fo gven 1, 2,, and β nvolvng mnmum sum of sk. He also studed attbute double samlng lan, Chs (0,1), Multle Defeed samlng lan of tye MDS(c 1,c 2 ) and MDS-1(c 1,c 2 ), RGS lan, and Lnk samlng lan nvolvng mnmum sum of oduce s and consume s sk. Raju (1984) has gven a set of tables fo fndng values ndexed though AQL and LQL fo fxed samle sze mnmzng + β wth and wthout weght fo Chs-1 lan. Raju (1984) has also followed Golub s aoach fo desgnng Multle Defeed state samlng lan of tye MDS-1 (c 1, c 2 ). Govndaaju and Subaman(1990) have studed the selecton of sngle samlng attbute lan nvolvng the mnmum sum of sks wthout fxng the samle sze osson model. Soundaaajan and Vjayaaghavan (1989b) have aled Golub s aoach fo desgnng Multle Defeed State Samlng MDS-1(0,2) lans nvolvng mnmum sks. Svasanka(2004) has desgned the secal tye double samlng lan nvolvng mnmum sum of sk and Two-Plan System TPS (c N, c T, n) nvolvng mnmum sum of sks. Chntha Zachaas (2006) has studed the selecton of Two-lan system TPS (n,kn;c) usng mnmum sum of sks. In accetance samlng, the oduce and consume lay a domnant ole and hence one allows cetan level of sks fo oduce and consume, namely =0.05 and β= Futhe ths aoach esults n the ounded values of 2 / 1. The exesson fo the sum of oduce s and consume s sk. +β=[1-p a ( 1 )]+P a ( 2 ) (1) If the oeatng ato 2 / 1 and n 1 ae known, then n 2 can be calculated as n 2 =( 2 / 1 )(n 1 ) (2) 2. Genealzed Two-Plan System Genealzed Two-Plan System whch s analogous of Dodge (1959) two-lan accetance samlng system. Dodge (1965), Stehens and Lason (1967), Calvn(1977), Hobbs(1987)etc have dscussed two-lan system n detal n vaous eseach aes. All the above dscussed swtchng ules can be vewed as a unque system of ules and can be genealzed. Kualman(1992) has ntoduced the swtchng ocedue fo such a genealzed two-lan system, and ts OC and ASN functons. The selecton of sngle samlng two-lan system wth equal samle szes but wth dffeent accetance numbes desgnated as TPS (n,c N,c T ). 2.1 Oeatng Pocedue Swtchng ules fo genealzed Two-lan Systems ae: Nomal to Tghtened When nomal nsecton s n effect, tghtened nsecton shall be nsttuted when s out of m consecutve lots o batches have been ejected on ognal nsecton (s m). Tghtened to Nomal When tghtened nsecton s n effect, nomal nsecton shall be nsttuted when d consecutve lots o batches have been consdeed accetable on ognal nsecton. A numbe of motant measues of efomance ae to be detemned and used n the evaluaton of OC functon whch wll be dscussed. P N = the ooton of lots exected to be acceted unde nomal nsecton. P T = the ooton of lots exected to be acceted unde tghtened nsecton. I N = the exected ooton of lots nsected on nomal nsecton. I T = the exected ooton of lots nsected on tghtened nsecton. Dodge (1959) has ovded a efomance measue wth a comoste of functon fo the obablty of accetance, P a () = I N P N + I T P T (3) Coyght 2014, Stateson Publcatons, Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9 Issue

3 Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9, Issue 1, The methods fo devng vaous measues of efomance fo the Genealzed Two- Plan System ae also studed. All obabltes can now be evaluated usng the condton that the sum of all obabltes equals to one,.e. I N + I T =1 (4) one can get, I N = I T = µ τ Whee, s 2 m s (1 a) (2a a 1) µ = (7) m s+ 1 s 1 a(1 a )(1 a) = the aveage numbe of lots nsected usng the nomal lan befoe gong to tghtened nsecton. d 1 b andτ = (8) d (1 b) b = the aveage numbe of lots nsected usng the nomal lan befoe gong to tghtened nsecton. Hee, a as P N and b as P T, the comoste OC and ASN functons ae, esectvely, obtaned as µ P τ Pa()= N + P T (9) Whee, P N = Pobablty of accetance unde the nomal nsecton. P N = (d c N /n,) P T = Pobablty of accetance unde the tghtened nsecton. P T = (d c T /n,) Note that whee µ and τ ae the aveage numbe of lots nsected usng nomal nsecton befoe gong to tghtened nsecton and aveage numbe of lots nsected usng tghtened nsecton befoe gong to nomal nsecton esectvely. 3. Condtons fo RDS Plan 1. Poducton s steady so that esult of ast, cuent and futue lots ae boadly ndcatve of a contnung ocess. 2. Lots ae submtted substantally n the ode of the oducton. 3. A fxed samle sze, n fom each lot s assumed. 4. Insecton s by attbutes wth qualty defned as facton non-confomng. 3.1 Oeatng Pocedue fo RDS Plan 1. Daw a andom samle of sze n fom the lot and detemne the numbe of defectves (d) found theen. 2. Accet the lot f d c 1.Reject the lot f d c 2. (5) (6) 3. If c 1 d c 2, accet the lot ovded oceedng o succeedng lots ae acceted unde RDS nsecton lan, othewse eject the lot. Hee c 1 and c 2 ae accetance numbes such that c 1 c 2, when =1 ths lan educes to RDS lan. 3.2 Oeatng Chaactestc fo RDS Plan The oeatng chaactestc functon P a () fo Reettve Defeed Samlng Plan s deved by Shanka and Mohaata (1991) usng the Posson Model as, a P a () = a(1 c ) + c (10) (1 c) whee c1 x a = [d c 1 ] = (11) c2 x c1 x c =[c 1 d c 2 ]= (12) also x=n. Thus the RDS lan s chaactezed wth aametes namely n, c 1, c 2 and the accetance cteon. 4. Selecton of Mnmum Rsk fo RDS Plan Table 1 s used to select a mnmum sks genealzed two-lan system fo gven 1 and 2. Fo the system of table, the oduce s and consume s sks wll be at most 10% each. Aganst the fxed value of the oeatng ato 2 / 1. Table 1 gve the accetance numbe c N (nomal accetance numbe), c T (tghtened accetance numbe) and the assocated oduce s and consume s sks n the body of the table aganst the oduct of samle sze and Accetable Qualty Level (n 1 ). The followng ocedue s used fo selectng the system fo gven 1, 2,, β. 1. Comutng the oeatng ato 2 / Wth the comuted value of 2 / 1, ente Table 1 n the ow headed by 2 / 1 whch s equal to o just smalle than the comuted ato. 3. The nomal accetance numbe c N, and the tghtened accetance numbe c T, ae obtaned when one oceeds fom left to ght n the ow dentfed n ste 2 such that the tabulated oduce s and consume s sks ae equal to just less than the desed values. 4. The samle sze n s obtaned as n=n 1 / 1. Whee n 1 values ae gven n the column headng coesondng to the accetance numbes obtaned n ste 3. Fo examle, f one fxes 1 = 0.02, 2 = 0.56 and = 0.05 and β = 0.10, fom table 1, one obtans a Two-Plan system TPS (n; c N, c T ) wth Reettve Defeed Samlng (RDS) lan as efeence lan nvolvng mnmum sum of sks as follows, 1. 2 / 1 = Tabulated 2 / 1 = 28.0 Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9 Issue 1 Page 36

4 K. K. Suesh, K. Vntha Xave 3. Coesondng to c N = 0, c T =1, gven n the body of the table, one obtans = 0.03, β = aganst the desed value of = 0.05, β = n = n 1/ 1 = 0.4/0.02 = Constucton of Tables The exesson fo the OC functon of Genealzed Two Plan system wth Reettve Defeed Samlng (RDS) Plan as efeence lan s gven by, P a () = µ P N + τp T (13) a P N = a(1 c) + c (1 c ) whee u1 x a = [d u 1 ] = c =[u 1 d u 2 ] = P T = 0! c u 2 a(1 c ) + c (1 ) v1 a = [d v 1 ] = e x u1 e 0! x a ( x) x ( x) (14) (15) v2 x v1 x c =[v 1 d v 2 ]= whee x=n s the OC functon of RDS lan as efeence lan. The exesson fo the sum of oduce s and consume s sk s gven as, + β = [1-P a ( 1 )] + P a ( 2 ) Fo fxed n 1 the value of n 2 s calculated fom equaton n 2 = ( 2 / 1 )(n 1 ) and s used n equaton + β = [1-P a ( 1 )] + P a ( 2 ). The aametes c N, c T coesondng to the mnmum [1-P a ( 1 )]+P a ( 2 ) ae obtaned by seachng fo c N = 0(0.03)20, c T = 1(0.041)0.40 wth the hel of a comute ogam. The values n table 1 gves oduce and consume sks whch ae obtaned coesondng to the values of u 1, u 2, v 1, v 2, s, m, d, c N, c T and fo whch the sum of sks s mnmum. 6. Concluson In accetance samlng, the oduce and consume lays a domnant ole and hence one allows cetan level of sks fo oduce and consume, namely =0.05 and β= In actce t s desable to desgn any samlng lan wth the assocated qualty levels whch concen to oduce and consume. Genealzed Two-Plan System and Reettve Defeed Samlng Plan have wde otental alcablty n ndustes to ensue hghe standad of qualty attanment and nceased custome satsfacton. Hee the selecton ocedues ae consdeed n ths ae usng the qualty levels assocated wth mnmum sum of oduce s and consume s sks wthout secfyng the samle sze athe than the fxed sks. Tables ae ovded n ths ae ae talo-made whch ae useful to the sho floo condton. Table 1: Paametc Values fo Two Plan System wth Reettve Defeed Samlng Plan usng Mnmum Sum of Rsks n 1 OR ,.5 1,.7 1,.8 2,.2 2,.3 2,.5 2,.7 2,.8 3,.2 3, ,.2 2,.4 2,.5 2,.7 2,.8 2,.9 3,.1 3,.2 3,.4 3, ,.2 2,.2 2,.3 2,.4 2,.5 2,.5 2,.6 2,.7 2,.8 2, ,.3 4,.4 4,.5 4,.6 4,.7 4,.8 3,2 5,.1 5,.4 5, ,.8 5,0 5,.1 5,.2 5,.3 5,.4 5,.5 5,.6 5,.8 5, ,.2 2,.2 2,.2 2,.2 2,.3 2,.3 2,.3 2,.3 2,.4 2,.4 2, ,.7 3,.7 3,.7 3,.7 3,.8 3,.8 3,.8 3,.9 3, ,.9 3,.9 3,.9 3,.9 2,2 2,2 2,2 2,2 4,.1 4,.1 4, ,.5 4,.5 4,.6 4,.6 4,.6 4,.6 4,.6 4,.7 4,.7 4,.7 4, ,.8 4,.8 4,.8 4,.8 4,.8 4, ,.8 5,.8 5,.8 5,.8 5,.8 5,.8 5,.9 5,.9 5,.9 5,.9 5, ,.3 6,.3 6,.3 6,.3 6,.3 6,.4 6,.4 6,.4 6,.4 6,.4 6,.5 Coyght 2014, Stateson Publcatons, Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9 Issue

5 Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9, Issue 1, ,6.4 3,3.9 4,8.4 3,3.9 4,8.4 2,4.9 6,6.4.9,6 Refeences 1. Calvn,T.W., TNT Zeo Accetance Numbe Samlng, Amecan Socety fo Qualty Contol, Techncal Confeence Tansactons, Phladelha, Pennsylvena,.35-39, Chntha Zachaas., Cetan Study Relatng to Genealzed Two-Plan System Indexed though Secfed Qualty Levels, MPhl Thess, Bhaatha Unvesty, Inda, Dodge, H.F., A Geneal Pocedue fo Samlng Insecton by Attbutes- Based on the AQL Concet, Techncal Reot No.10, The Statstcs Cente, Rutges The State Unvesty, New Bunswck, New Jesy, Dodge,H.F., Evoluton of Samlng Insecton System havng ules fo Swtchng between Nomal and Tghtened Insecton, Techncal Reot No.14, The Statstcs Cente, Rutges- The State Unvesty, New Bunswck, New Jesy, Dodge,H.F., A new Dual System of Accetance Samlng Insecton, Techncal Reot No.16, The Statstcs Cente, Rutges- The State Unvesty, New Bunswck, New Jesy, Govndaaju,K., Subaman,K., Selecton of Multle Defeed State MDS-1Samlng Plans fo Gven Accetable Qualty Level and Lmtng Qualty Level nvolvng Mnmum Rsks, Jounal of Aled Statstcs, Vol.17,No.3, , Golub,A., Desgnng Sngle Samlng Insecton Plans when the samle sze s fxed, J. Ame Statst Assn., Vol , Hald.A, Thyegod,P., The Comoste Oeatng Chaactestc Unde Nomal and Tghtened Samlng Insecton by Attbutes, Bulletn of the Intenatonal Statstcal Insttute, Vol.41, , Hobbs,C.F., Insecton by attbutes-a Method of Comang Rules fo Passng between Nomal and.1 2,6.4 6,6.4 2,10.4 2, ,3.5 Key C N C T % β%.1 9,.1 2,6.4 4,8.4 6,.9 7,.1 1,7.4 10,2.4 2,4.9 7,.1 12,.4 3,3.9 7,.1 2, ,4.5 8,4.4 2,4.9 7,.2 2, ,4.5 5,7.4 6,.9 Tghtened Insecton, Intenatonal Jounal of Qualty and Relablty Management, Vol.4, No.4,.47-60, Kualman,V., Studes on Desgnng Mnmum Insecton Attbute Accetance Samlng Plans, PhD Thess, Bhaatha Unvesty, Inda, Llly Chstna,A., Contbuton to the study of Desgn and Analyss of Susenson System and Some othe Samlng Plans, PhD Thess, Bhaatha Unvesty, Inda, MIL-STD-105D., Samlng Pocedues and Tables fo Insecton by Attbutes, US Govenment Pntng Offce, Washngton, DC, MIL -STD- 105E., Samlng Pocedues and Tables fo Insecton by Attbutes, US Govenment Pntng Offce, Washngton, DC, Raju,C., Contbutons to the study of Chan Samlng Plans, PhD Thess, Bhaatha Unvesty, Inda, Rombosk,L.D., An Investgaton of Quck Swtchng Accetance Samlng Systems, PhD Thess, Rutges- The State Unvesty, New Bunswck, New Jesey, Shanka,G., Mohaata,B.N., Analyss of Reettve Defeed Samlng Plans, IAPQR Tansactons, Vol.16, No.2,.17-25, Suesh,K.K., A study on Accetance Samlng Plan usng Accetable and Lmtng Qualty Levels, PhD Thess, Bhaatha Unvesty, Inda, Suesh,K.K., Samnathan,R., Constucton and Selecton of Reettve Defeed Samlng (RDS) Plan though Accetable and Lmtng Qualty Levels, Intenatonal Jounal of Statstcs and Systems, Vol.65, No.3, , Soundaaajan,V., Pocedues and Tables fo Constucton and Selecton of Chan Samlng Plans (ChS -1), Pat 1, Jounal of Qualty Technology, Vol.10, No.2,.56-60,1978a. Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9 Issue 1 Page 38

6 K. K. Suesh, K. Vntha Xave 20. Soundaaajan,V., Pocedues and Tables fo Constucton and Selecton of Chan Samlng Plans (ChS -1), Pat 2, Jounal of Qualty Technology, Vol.10, No.3, , 1978b. 21. Soundaaajan,V., Sngle Samlng Attbute Plans Indexed by AQL and AOQL, Jounal of Qualty Technology, Vol.13, No.3, , Soundaaajan,V., Govndaaju,K., A Note On Desgnng Chan Samlng Plans ChSP-1, The QR Jounal, Vol.9, No.3, , V.Soundaaajan,V., Vjayaaghavan,R., On Desgnng Multle Defeed State Samlng [MDS-1(0,2)] Plans Involvng Mnmum Rsks, Jounal of Aled Statstcs, Vol.16, No.1,.87-94,1989b. 24. Svasanka,S., Cetan esults on STDS lan and Genealzed Two-Plan System, MPhl Thess, Bhaatha Unvesty, Inda, Subaman,K., Studes on Desgnng Attbute Accetance Samlng Plans wth Emhass on Chan Samlng lans, PhD Thess, Bhaatha Unvesty, Inda, Vaest,R., A Pocedue to Constuct Multle Defeed State Samlng Plans, Methods of Oeatons Reseach, Vol.37, , Wotham,A.W., Bake,R.C., Multle Defeed State Samlng Insecton, The Intenatonal Jounal of Poducton Reseach, Vol.14, No.6, ,1976. Coyght 2014, Stateson Publcatons, Intenatonal Jounal of Statstka and Mathematka, ISSN: E-ISSN: , Volume 9 Issue