SELECTION OF MIXED SAMPLING PLANS WITH CONDITIONAL DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROUGH MAPD AND LQL USING IRPD

Transcription

1 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: SELECTION OF MIXED SAMPLING PLANS WITH CONDITIONAL DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROUGH AND LQL USING IRPD R. Samath Kumar 1, R. Vaya Kumar and R. Radhakrshnan 1. Assstant Professor, Deartment of Statstcs, Government Arts College, Combatore Assstant Professor, Deartment of Statstcs, SSM College of Arts and Scence,. Assocate Professor, Deartment of Statstcs, PSG College of Arts and Scence, Combatore 14. Abstract Ths aer resents the rocedure for the constructon and selecton of mxed samlng lan (MSP) usng Intervened Random effect Posson Dstrbuton (IRPD) as a baselne dstrbuton. Havng the condtonal double samlng lan as attrbute lan, the lans are constructed through lmtng qualty level (LQL) and maxmum allowable ercent defectve (). Tables are constructed for easy selecton of the lan. Key Words And Phrases: ntervened random effect osson dstrbuton, lmtng qualty level, mxed samlng lan, maxmum allowable ercent defectve, oeratng characterstc, osson, ntervened random effect osson dstrbuton. AMS (000) Subect Classfcaton Number: Prmary: 6P0 Secondary: 6D05 1. Introducton Mxed samlng lans consst of two stages of rather dfferent nature. Durng the frst stage the gven lot s consdered as a samle from the resectve roducton rocess and a crteron by varables s used to check rocess qualty. If rocess qualty s udged to be suffcently good, the lot s acceted. Otherwse the second stage of the samlng lan s entered and lot qualty s checked drectly by means of an attrbute samlng lan. The mxed samlng lans have been desgned under two cases of sgnfcant nterest. In the frst case, the samle sze n 1 s fxed and a ont on the OC curve s gven. In the second case, lans are desgned when two onts on the OC curve are gven. There are two tyes of mxed samlng lans called ndeendent and deendent lans. If the frst stage samle results are not utlzed n the second stage, then the lan s sad to be ndeendent otherwse deendent. The rncal advantage of mxed samlng lan over ure attrbute samlng lan s a reducton n samle sze for a smlar amount of rotecton. Schllng (196) roosed a method for determnng the oeratng characterstcs of mxed varables attrbutes samlng lans, sngle sded secfcaton and standard devaton known usng the normal aroxmaton. Baker and Brobst (198) have ntroduced the Condtonal Double Samlng Plan rocedures. It has Oeratng Characterstc Curves dentcal to those of comarable Double Samlng rocedures when the second samle s requred to make a decson, t can be obtaned from a related lot and not from the current lot. Condtonal Double Samlng Plan by usng samle nformaton from related lot results n more attractve Oeratng Characterstc Curves and smaller samle szes. Ths reducton n samle sze s the Prncal advantage of these rocedures over tradtonal samlng rocedures. Devaarul (00) has studed the mxed samlng lans and relablty based samlng lans. Radhakrshnan and Samath Kumar (006, 00 and 009) have constructed the mxed samlng lans usng Posson dstrbuton as a baselne dstrbuton. Samath Kumar (00) has constructed mxed varables attrbutes samlng lans ndexed through varous arameters. In the roduct control, the defectve unts are ether rebult or relaced by new unts durng the samlng erod. Qualty engneers are always nterested n mrovng the qualty level of roduct to enhance the satsfacton of the customers and hence, they kee makng changes n the roducton rocess. These actons trgger a change n the exected ncdence of defectve tems n the remanng observatonal erod. Any acton for reducng the number of defectves durng the samlng erod s called an nterventon and such nterventon arameter ranges from 0 to 1. In Intervened Random effect Posson Dstrbuton (IRPD), Posson arameter s modfed n two ways: one method s multlyng an nterventon arameter ρ (a constant) and secondly, multlyng an unobserved random effect whch follows Gamma robablty dstrbuton. The IRPD can be very useful to the qualty and relablty engneers, who always make changes n the roducton system n the observatonal erod of qualty checkng to ensure relablty of the system, because, the falure rate of the comonents may vary n dfferent tme ntervals. The other areas of alcaton of IRPD are queung, demograhc studes and rocess control and so on. Shanmugam (1985) has used Intervened Posson Dstrbuton (IPD) n the lace of Zero Truncated Posson Dstrbuton (ZTPD) for the study on cholera cases. Radhakrshnan and Sekkzhar (00a, b, c) ntroduced Intervened Random effect Posson Dstrbuton n the lace of Posson dstrbuton for the constructon of attrbute samlng lans. IJCER Mar-Ar 01 Vol. Issue No Page 06

2 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: In ths aer, usng the oeratng rocedure of mxed samlng lan (ndeendent case) wth condtonal double samlng lan as attrbute lan, tables are constructed usng IRPD as a baselne dstrbuton. The tables are constructed for mxed samlng lan (MSP) ndexed through ) LQL ). The lan ndexed through s comared wth the lan ndexed through LQL.. Condtons For Alcatons Of IRPD - Mxed Samlng Plan Producton rocess s modfed durng the samlng nsecton by an nterventon. Lots are submtted substantally n the order of ther roducton. Insecton s by varable n the frst stage and attrbute n the second stage wth qualty defned as the fracton defectve. Lot qualty varaton exsts.. Glossary of symbols: The symbols used n ths aer are as follows: : submtted qualty of lot or rocess Pa ( ): robablty of accetance for gven qualty : submtted qualty such that P a ( ) = 0.10 (also called LQL) : maxmum allowable ercent defectve () n : samle sze for each lot n 1,1 : frst samle sze for varable samlng lan n 1, : frst samle sze for attrbute samlng lan n, : second samle sze for attrbute samlng lan c 1 : frst attrbutes accetance number c : second attrbutes accetance number c : thrd attrbutes accetance number d : number of defectves n the th samle (=1,,,..) : robablty of accetance for the lot qualty : robablty of accetance under varables lan for ercent defectve sze n 1 ) : robablty of accetance under attrbutes lan for ercent defectve z () sze n ) : z value for the th ordered observaton k : varable factor such that a lot s acceted f X L k (wth samle (wth samle 4. Oeratng Procedure Of Mxed Samlng Plan Havng Condtonal Double Samlng Plan As Attrbute Plan Schllng (196) has gven the followng rocedure for the ndeendent mxed samlng lan wth lower secfcaton lmt (L) and standard devaton ( ). Determne the arameters of the mxed samlng lan n 1, 1, n 1,, n,, k, c 1, c and c Take a random samle of sze n 1,1 from the lot If a samle average X L k, accet the lot If a samle average X < L k take a second samle of sze n 1, ( e., n 1, =n 1, 1 ) Insect all the artcles ncluded n the samle. Led d be the number of defectves n the samle If d c 1, accet the lot If d >c, reect the lot If c 1 +1 d c, then take a second samle of sze n, from the recedng (-1) lot or the next (+1) lot Fnd the number of defectves d -1 or d +1. Then fnd d= d +d -1 or d=d +d +1 If d c, accet the lot otherwse reect the lot. 5. Constructon of mxed samlng lan havng condtonal double samlng lan as attrbute lan usng rd. Schllng (196) has gven the OC functon of mxed samlng lan as IJCER Mar-Ar 01 Vol. Issue No Page 0

3 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: c L ( ) = Pn 1 ( X A) + Pn 1 ( X > A) ; n (1) The above exresson s gven as = 0 + (1- ) () The oeraton of mxed samlng lans can be roerly assessed by the OC curve for gven values of the fracton defectve. The develoment of mxed samlng lans and the subsequent dscussons are lmted only to the uer secfcaton lmt U. By symmetry, a arallel dscusson can be made for lower secfcaton lmts. The rocedure for the constructon of mxed varables attrbutes samlng lans s rovded by Schllng (196) for a gven n 1,1, k and a ont on the OC curve s gven below. Assume that the mxed samlng lans are ndeendent Slt the robablty of accetance ( ) determnng the robablty of accetance that wll be assgned to the frst stage. Let t be Decde the samle sze n 1,1 (for varable samlng lan) to be used Calculate the accetance lmt for the varable samlng lan as L k L [ z( ) { z( ) / n }], where L s the lower secfcaton lmt and z (t) s the standard normal varate corresondng to t such that t = 1,1 zt () 1 u / e Determne the samle average X. If a samle average X < L k, take a second stage samle sze n 1, usng attrbute samlng lan. Slt the robablty of accetance as and, such that = du + (1- ) and fx the value of. Now determne, the robablty of accetance assgned to the attrbutes lan assocated wth the second stage samle as =( - )/(1- ) Determne the arorate second stage samle sze n 1, from Pa ( ) = for = Usng the above rocedure, tables can be constructed to facltate easy selecton of mxed samlng lan wth condtonal double samlng lan as attrbute lan usng IRPD as a baselne dstrbuton ndexed through and LQL. Radhakrshnan and Sekkzhar (00a, b and c) suggested the robablty mass functon of the CDSP usng IRPD as a baselne dstrbuton for n=n 1, =n, s Pa ( ) = where c1 c c1 1 c c1 c c q q... q () c1 1 c1 c q = = 1! l m e m l 1 m l0 1 m l! l! 1! l e l 1! 1 l0 1 l! l! 1! and = The tables furnshed n ths aer are for the case when α=1, m= and n=n 1, =n,. IJCER Mar-Ar 01 Vol. Issue No Page 08

4 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: Constructon Of Mxed Samlng Plans Indexed Through And MAAOQ, ntroduced by Mayer (196) and studed by Soundararaan (195) s the qualty level corresondng to the nflecton ont of the OC curve. The degree of sharness of nsecton about ths qualty level s measured by t, the ont at whch the tangent to the OC curve at the nflecton ont cuts the roorton defectve axs for desgnng, Soundararaan (195) roosed a selecton rocedure for sngle samlng lan ndexed wth and K= t. Usng the robablty mass functon of the IRPD, gven n exresson (), the nflecton ont ( ) s obtaned by usng d Pa ( ) d Pa ( ) = 0 and 0. The n 1, values are calculated for dfferent values of c 1, c, c and ρ=0. for 0.04 usng d d c++ rogram and resented n Table 1. The MAAOQ (Maxmum Allowable Average Outgong Qualty) of a samlng lan s defned as the Average Outgong Qualty (AOQ) at the. By defnton AOQ = P ( ) and R MAAOQ = a Pa ( ) The values of and MAAOQ are calculated for dfferent values of c 1, c, c and ρ for 0.0 and the rato MAAOQ s resented n Table 1. Table 1: n 1, and n 1, MAAOQ values for a secfed values of c 1, c, c and dfferent values of ρ for mxed samlng lan when 0.04 ρ c 1 c c n 1, n 1, MAAOQ R MAAOQ IJCER Mar-Ar 01 Vol. Issue No Page 09

5 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: Selecton of the lan For the gven values of ρ, MAAOQ, and MAAOQ, the rato R s found and the nearest value of R s located n Table 1. The corresondng value of c 1, c, c and n 1, values are noted and the value of n, s obtaned usng n 1, n1,. Examle 1: Gven ρ=0.8, 0.04, =0.09 and MAAOQ=0.041, the rato MAAOQ R = s 0.09 comuted. In Table 1 the nearest R value s whch s corresondng to c 1 =, c =6, c =1. The value of n 1, =.689 s n1,.689 found and hence the value of n 1, s determned as n1, 40. Thus n , =40, n, =0, c 1 =, c =6 and c =1 are the arameters of mxed samlng lan havng CDSP as attrbute lan usng IRPD as a baselne dstrbuton for the gven values of ρ=0.8, =0.09 and MAAOQ= Practcal roblem: Suose the lan n 1,1 =1, k=g s the lot by lot accetance nsecton of a health drnk roduct wth carbohydrate secfcaton 6g ( 500g ack) wth known S.D(σ)=1.5g. In ths examle L=6g, σ =1.5g and k=g, L k = 6 + (1.5)=64.5g Now by alyng the varables nsectng frst, take random samle of sze n 1,1 =1 from the lot. Record the samle results and fnd X. If X L k =64.5g, then accet the lot. If X <, take a random samle of sze n 1, and aly the attrbute nsecton. Under attrbutes nsecton, by usng Condtonal double samlng lan as attrbute lan usng ntervened Random effect Posson Dstrbuton (IRPD) as a baselne dstrbuton, f the manufacturer fxes the values = 0.09(9. non conformtes out of 100), MAAOQ=0.041(41 non conformtes out of 100) and 0.04, take a samle of sze n 1, =40 and observe the number of defectves d. If d, accet the lot and f d >6, reect the lot. If 4 d 6, take a second samle of sze n, =0 from the remanng lot and fnd the number of defectves (d). If d 1 accet the lot otherwse reect the lot and nform the management for further acton.. Constructon Of Mxed Samlng Plans Indexed Through LQL The rocedure gven n secton 5 s used for constructng the mxed samlng lan ndexed through LQL ( ). By assumng the robablty of accetance of the lot be =0.10 and ρ usng c++ rogram and s resented n Table. =0.14, the n 1, values are calculated for dfferent values of c 1, c, c and IJCER Mar-Ar 01 Vol. Issue No Page 10

6 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: Table : n 1, LQL values for a secfed values of c 1, c, c and ρ of mxed samlng lan when =0.10 and =0.04 ρvalues Selecton of the lan Table s used to construct the lans when, ρ, c 1, c and c are gven. For any gven values of, ρ, c 1, c and c one can n1, determne n 1, value usng n1,. n1, Examle : Gven ρ=0., = , c 1 =, c =, c =1 and.56 =0.04. Usng Table, fnd n1, For a fxed 0.04, the mxed samlng lan wth CDSP as attrbute lan s n 1, =, n, =6, ρ=0., c 1 =, c = and c =1. 8. Comarson Of Mxed Samlng Plan Indexed Through And LQL In ths secton MSP ndexed through s comared wth MSP ndexed through IQL by fxng the arameters c 1, c, c and For the secfed values of ρ, and MAAOQ wth the assumton for 0.04 one can fnd the values of c 1, c and c ndexed through. By fxng the values of c 1, c and c fnd the value of by equatng P ( ) = =0.10. For =0.04, c1, c and c one can fnd the values of n, usng IJCER Mar-Ar 01 Vol. Issue No Page 11 n n a. 1, 1, from Table. For dfferent combnatons of ρ, and MAAOQ the values of c 1, c, c and n 1, (ndexed through ) and c 1, c, c and n 1, (ndexed through LQL) are calculated and resented n Table.

7 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: Constructon of OC curve The OC curves for the lan ρ=0.8, n 1, =40, n, =0, c 1 =, c =6, c =1 (ndexed through ) and n 1, =44, n, =, c 1 =, c =6, c =1 (ndexed through LQL) based on the dfferent values of n1, and Pa ( ) are resented n Fgure 1. Table : Comarson of the Plans Gven Values Indexed Through Indexed Through LQL MAAOQ ρ c 1 c c n 1, n, c 1 c c n 1, n, OC curves are drawn Fg1: OC curves for the lans (ρ=0.8, c 1 =, c =6, c =1, n 1, =40, n, =0) and (ρ=0.8, c 1 =, c =6, c =1, n 1, =44, n, =) 9. Concluson In ths aer the constructon of mxed samlng lan wth condtonal double samlng lan as attrbute lan ndexed through the arameters and LQL are resented by takng IRPD as a baselne dstrbuton. Further the lan ndexed through s comared wth the lan ndexed through LQL. It s concluded from the study that the second stage samle sze requred for condtonal double samlng lan ndexed through s less than that of second stage samle sze of the condtonal double samlng lan ndexed through LQL. If the floor engneers know the levels of or LQL, they can have ther samlng lans on the floor tself by referrng to the tables. Ths rovdes the flexblty to the floor engneers n decdng ther samlng lans. Varous lans can also be constructed to make the system user frendly by changng the frst stage robabltes ( also be comared for ther effcency., ) and can IJCER Mar-Ar 01 Vol. Issue No Page 1

8 R. Samath Kumar, R. Vaya Kumar, R. Radhakrshnan /Internatonal Journal Of Comutatonal Engneerng Research / ISSN: References 1. R.C. Baker, and R.W. Brobst, 198, Condtonal Double Samlng Plan, Journal of Qualty Technology, Vol. 10, No. 4, Devaarul, S., 00, Certan Studes Relatng to Mxed Samlng Plans and Relablty Based Samlng Plans, Ph.D., Dssertaton, Bharathar Unversty, Combatore, Taml Nadu, Inda.. P.L. Mayer, 196, A note on sum of Posson robabltes and an alcaton, Annals of Insttute of Statstcal Mathematcs, Vol.19, R. Radhakrshnan, and R. Samath Kumar, 006, Constructon of mxed samlng lan ndexed through and AQL wth chan samlng lan as attrbute lan, STARS, Vol., No.1, June 006, R. Radhakrshnan, and R. Samath Kumar, 00, Constructon and comarson of mxed samlng lans havng reettve grou samlng lan as attrbute lan, Natonal Journal of Technology, Vol. 4, No R. Radhakrshnan, and R. Samath Kumar, 009, Constructon and comarson of mxed samlng lans havng ChSP-(0,1) lan as attrbute lan, The Internatonal Journal of Statstcs and Management System, Vol.4, No.1-, R. Radhakrshnan, and J. Sekkzhar, 00a, Constructon of samlng lans usng ntervened random effect Posson dstrbuton. The Internatonal Journal of Statstcs and Management Systems, Vol., 1-, R. Radhakrshnan, and J. Sekkzhar, 00b, Constructon of condtonal double samlng lans usng ntervened random effect Posson dstrbuton, Proceedngs volume of SJYSDNS-005, Acharya Nagaruna Unversty, Guntur. P R. Radhakrshnan, and J. Sekkzhar, 00c, Alcaton of ntervened random effect Posson dstrbuton n rocess control lans, Internatonal Journal of Statstcs and Systems. Vol.. No Samath Kumar, R., 00, Constructon and Selecton of Mxed Varables Attrbutes Samlng Plans, Ph.D., Dssertaton, Bharathar Unversty, Combatore, Taml Nadu, Inda. 11. Schllng, E.G., 196, A General Method for Determnng the Oeratng Charaterstcs of Mxed Varables Attrbute Samlng Plans Sngle Sde Secfcatons, S.D. known, Ph.D Dssertaton Rutgers The State Unversty, New Brunswck, New Jersy. 1. R. Shanmugam, 1985, An ntervened Posson dstrbuton and ts medcal alcatons, Bometrcs, 41, V. Soundararaan, 195, Maxmum allowable ercent defectve () sngle samlng nsecton by attrbutes lan, Journal of Qualty Technology, Vol., No.4, IJCER Mar-Ar 01 Vol. Issue No Page 1