523 P a g e. is measured through p. should be slower for lesser values of p and faster for greater values of p. If we set p*

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1 R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: Vol., Issue 4, July-August 0, pp.5-58 Constructon Of Mxed Smplng Plns Indexed Through Mpd And Aql Wth Condtonl Double Smplng Pln As Attrbute Pln Usng Weghted Posson Dstrbuton R. Smpth Kumr, R. Kruthk nd R. Rdhkrshnn. Assstnt Professor, Deprtment of Sttstcs, Government Arts College, Combtore Assstnt Professor n Sttstcs, Klgnr Krunndh Insttute of Technology, Combtore Assocte Professor, Deprtment of Sttstcs, PSG College of Arts nd Scence, Combtore ABSTRACT Ths pper presents the procedure for the constructon nd selecton of mxed smplng pln wth MAPD s qulty stndrd nd Condtonl double smplng pln s ttrbute pln usng Weghted Posson dstrbuton s bse lne dstrbuton. The plns re constructed ndexed through MAPD nd AQL nd lso compred. Tbles re constructed for the esy selecton of the plns. Keywords nd Phrses: Mxmum Allowble Percent Defectve, Acceptble qulty level, Opertng Chrcterstc, Tngent ntercept. AMS (000) Subject Clssfcton Number: Prmry: 6P0 Secondry: 6D05. INTRODUCTION Mxed smplng plns consst of two stges of rther dfferent nture. Durng the frst stge the gven lot s consdered s smple from the respectve producton process nd crteron by vrbles s used to check process qulty. If process qulty s judged to be suffcently good, the lot s ccepted. Otherwse the second stge of the smplng pln s entered nd lot qulty s checked drectly by mens of n ttrbute smplng pln. There re two types of mxed smplng plns clled ndependent nd dependent plns. If the frst stge smple results re not utlzed n the second stge, then the pln s sd to be ndependent otherwse dependent. The prncpl dvntge of mxed smplng pln over pure ttrbute smplng plns s reducton n smple sze for smlr mount of protecton. It s the usul prctce tht whle selectng smplng nspecton pln, to fx the opertng chrcterstc (OC) curve n ccordnce wth the desred degree of dscrmnton. The smplng pln s n turn fxed through sutbly chosen prmeters. The entry prmeters used n the cceptnce smplng lterture re cceptble qulty level (AQL), ndfference qulty level (IQL), lmtng qulty level (LQL) nd mxmum llowble percent defectve (MAPD). Severl uthors hve provded procedures to desgn the smplng plns ndexed through these prmeters for vrous cceptnce smplng plns. The concept of MAPD ( p * ) ws ntroduced by Myer (967) nd further studed by Soundrrjn (975) s the qulty level correspondng to the nflecton pont on the OC curve. The degree of shrpness of nspecton bout ths qulty level p * s mesured through p t, the pont t whch the tngent to the OC curve t the nflecton pont cuts the proporton defectve xs. One of the desred propertes of n OC curve s tht the decrese of ( p) should be slower for lesser vlues of p nd fster for greter vlues of p. If we set p* s the qulty stndrd, the bove property of the OC curve s obtned snce p* corresponds to the nflecton pont of the OC curve nd hence d P ( p) / dp = 0 for p= p * P d P ( p) / dp < 0 for p< * d P ( p) / dp > 0 for p> p * The mxed smplng pln hs been desgned under two cses of sgnfcnt nterest. In the frst cse smple sze n s fxed nd pont on the OC curve s gven. In the second cse plns re desgned when two ponts on the OC curve re gven. The procedure for desgnng the mxed smplng plns to stsfy the bove mentoned condtons ws provded by Schllng (967). Usng Schllng s procedure, Devrul (00) hs constructed tbles for mxed p 5 P g e

2 R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: Vol., Issue 4, July-August 0, pp.5-58 smplng plns (ndependent cse) hvng vrous smplng plns s ttrbute plns. Rdhkrshnn nd Smpth Kumr (006, 006 b, 007, 009) nd Rdhkrshnn et.l (00) hve mde contrbutons to mxed smplng pln for ndependent cse. Smpth Kumr (007) hs constructed mxed smplng pln wth vrous smplng plns s ttrbute plns usng Posson dstrbuton s bse lne dstrbuton. Rdhkrshn Ro (977) suggested tht the Weghted Bnoml dstrbuton cn be used n desgnng smplng plns. Sudeswr (00) studed the constructon of smplng plns usng Weghted Posson dstrbuton s bse lne dstrbuton. Mohn Pry (008), Rdhkrshnn nd Mohn Pry (008, 008 b) hve constructed the smplng plns usng Weghted Posson dstrbuton s bse lne dstrbuton. The Weghted Posson dstrbuton plys n mportnt role n the cceptnce smplng, mnly n the constructon of smplng plns. Ech outcome (number of defectves) s specfc but cn be ssgned wth dfferent weghts bsed on ts mportnce or usge. In ths pper, mxed smplng pln (ndependent cse) wth Condtonl double smplng pln s ttrbute pln s constructed usng Weghted Posson dstrbuton s bse lne dstrbuton. The plns ndexed through MAPD nd AQL re constructed seprtely by fxng the vlues of c, c, c nd β j '. The mxed plns ndexed through MAPD (p * ) nd AQL (p ) re lso compred.. GLOSSARY OF SYMBOLS The symbols used n ths pper re s follows: p : submtted qulty of lot or process P (p) : probblty of cceptnce for gven qulty p : mxmum llowble percent defectve p * p t : the pont t whch the nflecton tngent of the OC curve cuts the p xs p : the submtted qulty level such tht P (p ) = 0.95 (lso clled AQL) h * : reltve slope t p * n n : smple sze for the vrble smplng pln : smple sze for the ttrbute smplng pln c : cceptnce number β j : probblty of cceptnce for the lot qulty p j β j ' : probblty of cceptnce ssgned to frst stge for percent defectve p j β j '' : probblty of cceptnce ssgned to second stge for percent defectve p j k X A = U - k : vrble fctor such tht lot s ccepted f. OPERATING PROCEDURE OF MIXED SAMPLING PLAN HAVING CONDITIONAL DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN Schllng (967) hs gven the followng procedure for the ndependent mxed smplng pln wth upper specfcton lmt (U) nd stndrd devton (σ).. Determne the prmeters of the mxed smplng pln n, n,, n,, k, c, c nd c.. Tke rndom smple of sze n from the lot.. If smple verge X A = U - k, ccept the lot 4. If the smple verge X > A = U - k, tke second smple of sze n, nd count the number of defectves d n the smple. 5. If the number of defectves d c, ccept the lot. 6. If c + d c, tke second smple of sze n, from the remnng lot nd fnd the number of defectves d 7. If d c or d +d c ccept the lot, otherwse reject the lot. 4. CONSTRUCTION OF MIXED SAMPLING PLAN HAVING CONDTIONAL DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN USING WEIGHTED POISSON DISTRIBUTION The detled procedure dopted n ths pper for the constructon of mxed smplng pln hvng Condtonl double smplng s ttrbute pln usng Weghted Posson dstrbuton ndexed through MAPD s gven below: Assume tht the mxed pln s ndependent Decde the smple sze n (for vrble smplng pln) to be used. Clculte the cceptnce lmt for the vrble smplng pln s A U [ z( p ' * ) { z( * ) / n }], where z s stndrd norml vrte correspondng to t such tht t = z( t) e u du Splt the probblty of cceptnce β * s β * ' nd β * ''such tht β * = β * ' + (- β * ') β * ''. Fx the vlue of β * ' 54 P g e

3 R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: Vol., Issue 4, July-August 0, pp.5-58 Determne β * '', the probblty of cceptnce P (p) = ssgned to the ttrbute pln ssocted wth the c c c c c second stge smple s β * ''= (β * - β * ')/ (- β * '). P Pc q Pc q... Determne the pproprte second stge smple of n from the relton β * '' = np e ( np) c c c c c c cwhere P P Where P c q P np e ( np) ( )! c, q q... P c e mnp ( mnp) ( )! Usng the bove procedure, tbles hve been constructed to fcltte esy selecton of mxed smplng pln usng Condtonl Double Smplng pln s ttrbute plns ndexed through MAPD. 4. Constructon of tbles The OC functon of Weghted Posson dstrbuton s gven by, P ( p) x x0 x p( x, ) p( x; ) ; x=0,, The probblty of cceptnce for Condtonl double smplng pln under Weghted Posson dstrbuton when α = s used n ths pper for determnng the second stge probbltes nd s gven by q q e P mnp, ( mnp) ( )! ( )!, m. c c P c In ths pper, the probblty mss functon of the condtonl double smplng pln s used for m=. For n, = n, = n (sy) the nflecton pont (p * ) s d p ( p) obtned by usng 0 dp d p ( p) nd 0. dp The reltve slope of the OC curve h * s gven by, h * = p P ( p) dp ( p) dp t p=p *. The nflecton tngent of the OC curve cuts the p xs t p t =p * + (p * /h * ). The vlues of n p *, h *, n p t nd R= p t / p * re clculted for the specfed β * ' = 0.5 usng C progrm nd presented n Tble. q Tble : Vrous chrcterstcs of the mxed smplng pln when (p *, β * ) nd (p, β ) re known for specfed β * ' = 0.5, β =0.95, β ' =0.5 c c c np β * β * '' np * h * np t R =p t /p * P g e

4 R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: Vol., Issue 4, July-August 0, pp Selecton of the Pln Tble s used to construct the plns when MAPD (p * ) nd tngent ntercept (p t ) re gven. For ny gven vlues of c,c, p t nd p * one cn fnd the rto R = p t / p *. Correspondng to the vlue of c nd c fnd the vlue n Tble under the column R whch s equl to or just greter thn the specfed rto, the correspondng vlue of c s noted from ths c,c nd c vlues one cn determne the vlue of n usng n = n p * / p *. Exmple : Gven c = 6, c = 0, p * = 0.09, p t =0. nd β * ' = 0.5. Fnd the rto R= p t /p * =.0. Usng Tble, correspondng to c = 6, c =0 select the vlue of R equl to or just greter thn ths rto s.67 whch s ssocted wth c = 6, c =0, c = nd n = n p * / p * = (7.66/0.09) = 85.The Mxed Smplng pln wth Condtonl double smplng pln s ttrbute pln s n, = 85,n, =85, c = 6, c =0 nd c =. Prctcl problem: Suppose the pln n =, k=.0 s to be ppled to the lot by lot cceptnce nspecton of lptop bttery, the chrcterstc to be nspected s the opertng temperture nd mbent temperture of the bttery for whch there s specfed upper lmt of F wth known S.D (σ) of.0 0 F. In ths exmple, U= F, σ=.0 0 F nd k=.0 A = U-kσ = 60 - (.0) (.0) = 60-4 =56 0 F. Now, by pplyng the vrble nspecton frst, tke rndom smple of sze n = from the lot. Record the smple results nd fnd X. If X A = U - k =56 0 F, then ccept the lot. If X >A, tke rndom smple sze n, nd pply the ttrbute nspecton. Under ttrbutes nspecton, by tkng Condtonl double smplng pln s ttrbute pln, f the mnufcturer fxes the vlues p * =0.09 (9 non conformtes out of 00), p t =0.( non conformtes out of 00) nd β * ' =0.5, tke the smple of sze n, =85 nd observe the number of defectves (d ). If 56 P g e

5 R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: Vol., Issue 4, July-August 0, pp.5-58 d 6 ccept the lot nd f d >, reject the lot. If 7 d, tke second smple of sze n, =85 from the remnng lot nd fnd the number of defectves (d ). If d 0 or d +d ccept the lot, otherwse reject the lot nd nform the Mngement for further cton. 5. SELECTION OF MIXED SAMPLING PLAN HAVING CONDITIONAL DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROGH AQL The generl procedure gven n secton 4 s used for desgnng the mxed smplng pln hvng Condtonl double smplng pln s ttrbute pln ndexed through AQL (p ). For n, = n, =n(sy),under the ssumpton β =0.95 nd β ' =0.5, the np vlues re clculted for dfferent vlues of c,c nd c usng C progrm nd presented n Tble. 5. Selecton of the Pln Tble s used to construct the plns when AQL (p ), c, c, c vlues re gven. For ny specfed vlues of p,c, c nd c one cn determne n vlue usng n= np / p. Exmple : Gven p = 0.0, c = 4,c = 7,c = 0 nd β ' =0.5. Usng Tble, fnd n=np /p = * OC Curves re drwn Tble : Comprson of plns c c Gven Vlues p * p t Through MAPD n,c.6594/0.0=89. For fxed β ' =0.5, the Mxed Smplng pln wth Condtonl double smplng pln s ttrbute pln s n, = 89, n, =89, c = 4, c =7 nd c =0. 6. COMPARISON OF PLANS INDEXED THROUGH MAPD AND AQL In ths secton the mxed smplng pln wth Condtonl double smplng pln s ttrbute pln ndexed through MAPD s compred wth the mxed smplng pln wth Condtonl double smplng pln s ttrbute pln ndexed through AQL by fxng the prmeters c,c nd β '. For the specfed vlues of p * nd p t, one cn fnd the vlues of c,c, c nd n ndexed through MAPD for β * ' = 0.5 s gven n secton 4. By fxng the vlues of c,c, c nd n, fnd the vlue of p by equtng P (p) = β = By fxng β ' =0.5 nd c,c, c nd for the vlue of p, one cn fnd the vlue of n usng n = np /p from Tble,for dfferent combntons of c,c, p * nd p t, the vlues of n, c (ndexed through MAPD) nd n, c (ndexed through AQL) re clculted nd presented n Tble. Through AQL n,c , 6 9, , 5 69, , 7, , 7 84, , * 9, * 57 P g e

6 R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: Vol., Issue 4, July-August 0, pp CONCLUSION In ths pper the procedure for constructng mxed smplng pln s ndexed through MAPD wth Weghted Posson dstrbuton s the bselne dstrbuton s presented. Sutble tbles re lso provded for the esy selecton of the plns for the engneers who re workng on the floor of the ssembly. It s concluded from the study tht the second smple sze requred for mxed smplng pln wth Condtonl double smplng pln s ttrbute pln ndexed through MAPD s less thn tht of the second stge smple sze of the Mxed smplng pln wth Condtonl double smplng pln s ttrbute pln ndexed through AQL, justfed by Smpth Kumr (008).These plns defntely help the producers, becuse of the lesser smple sze whch drectly result n lesser smplng cost nd ndrectly reduces the totl cost of the product. The dfferent smplng plns cn lso be constructed by chngng the frst stge probbltes (β * ' nd β ') nd cn be compred for ther effcency. REFERENCES [] Devrul, S., 00, Certn studes reltng to mxed smplng plns nd relblty bsed smplng plns, PhD Dssertton, Deprtment of Sttstcs, Bhrthr Unversty, Combtore, Tml Ndu, Ind. [] Myer, P.L., 967, A note on sum of Posson Probbltes nd n Applcton, Industrl Qulty Control, Vol 9, No.5, pp.-5. [] Mohn Pry, L., 008,Constructon nd Selecton of cceptnce smplng plns bsed on Weghted Posson dstrbuton, PhD Dssertton, Deprtment of Sttstcs, Bhrthr Unversty, Combtore, Tml Ndu, Ind. [4] Rdhkrshn Ro, C., 977, A nturl exmple of Weghted Bnoml dstrbuton, The Amercn Sttstcn, Vol,, No.. [5] Rdhkrshnn, R., nd Mohn Pry, L., 008, Selecton of sngle smplng pln usng Condtonl Weghted Posson dstrbuton, Interntonl Journl of Sttstcs nd Systems, Vol, No., pp [6] Rdhkrshnn, R., nd Mohn Pry, L., 008 b, Comprson of CRGS plns usng Posson nd Weghted Posson dstrbuton, Probstt Forum, Vol, No., pp [7] Rdhkrshnn, R., nd Smpth Kumr, R., 006, Constructon of mxed smplng pln ndexed through MAPD nd IQL wth sngle smplng pln s ttrbute pln, Ntonl Journl of Technology, Vol, No., pp [8] Rdhkrshnn, R., nd Smpth Kumr, R., 006 b, Constructon of mxed smplng plns ndexed through MAPD nd AQL wth chn smplng pln s ttrbute pln, STARS Int. Journl, Vol 7, No, pp. 4-. [9] Rdhkrshnn, R., nd Smpth Kumr, R., 007, Constructon of mxed smplng plns ndexed through MAPD nd IQL wth double smplng pln s ttrbute pln, The Journl of the Kerl Sttstcl Assocton, Vol 8, pp. -. [0] Rdhkrshnn, R., nd Smpth Kumr, R., 009,Constructon nd comprson of mxed smplng plns hvng ChSP-(0,) pln s ttrbute pln, The Interntonl Journl of Sttstcs nd Mngement System,Vol 4, No.-, pp [] Rdhkrshnn, R., Smpth Kumr, R., nd Mlth, M., 00, Selecton of mxed smplng pln wth TNT- (n, n ; 0) pln s ttrbute pln ndexed through MAPD nd MAAOQ, Interntonl Journl of Sttstcs nd System, Vol 5, No 4, pp [] Smpth Kumr, R., 007, Constructon nd selecton of mxed vrbles-ttrbutes smplng pln - PhD Dssertton, Deprtment of Sttstcs, Bhrthr Unversty, Combtore, Tml Ndu, Ind. [] Schllng, E.G., 967, A generl method for determnng the opertng chrcterstcs of mxed vrbles-ttrbutes smplng plns sngle sde specfctons, S.D.known, PhD Dssertton- Rutgers- The Stte Unversty, New Brunswck, New Jersy. [4] Soundrrjn, V., 975, Mxmum llowble percent defectve (MAPD) sngle smplng nspecton by ttrbute pln, Journl of Qulty Technology, Vol 7, No. 4, pp.7-8. [5] Sudeswr 00, Desgnng of smplng pln usng Weghted Posson dstrbuton, M.Phl. Dssertton, Deprtment of Sttstcs, PSG College of Arts nd Scence, Combtore. 58 P g e