CONSTRUCTION OF MIXED SAMPLING PLAN WITH DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROUGH (MAPD, MAAOQ) AND (MAPD, AOQL)

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1 Global J. of Arts & Mgmt., 0: () Researh Paer: Samath kumar et al., 0: P.07- CONSTRUCTION OF MIXED SAMPLING PLAN WITH DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROUGH (MAPD, MAAOQ) AND (MAPD, AOQL) Samath Kumar, R., Kiruthika, R and R. Radhakrishnan Det. of Statistis, Government Arts College, Coimbatore-6408, Tamil Nadu,, India. Det. of Statistis, Kalaignar Karunanidhi Institute of Tehnology, Coimbatore-6440, Tamilnadu, India. Det. of Statistis, PSG College of Arts and Siene, Coimbatore-6404, Tamilnadu, India. Corresonding author: rkrishnan_researh@yahoo.om ABSTRACT This aer resents the roedure for the onstrution and seletion of the mixed samling lan using MAPD as a quality standard and MAAOQ as a measure for outgoing quality with the double samling lan (DSP) as attribute lan using weighted Poisson distribution as a base line distribution. The lans indexed through (MAPD, MAAOQ) and (MAPD, AOQL) are onstruted and omared. Tables are rovided for the easy seletion of the lans. Keywords: Maximum allowable erent defetive, Maximum allowable average outgoing quality level, Average outgoing quality level, Oerating Charateristi. INTRODUCTION Mixed samling lans onsist of two stages of rather different nature. During the first stage the given lot is onsidered as a samle from the resetive rodution roess and a riterion by variables is used to hek roess quality. If roess quality is judged to be suffiiently good, the lot is aeted. Otherwise the seond stage of the samling lan is entered and lot quality is heked diretly by means of an attribute samling lan. There are two tyes of mixed samling lans alled indeendent and deendent lans. If the first stage samle results are not utilized in the seond stage, then the lan is said to be indeendent otherwise deendent. The rinial advantage of a mixed samling lan over ure attribute samling lans is a redution in samle size for a similar amount of rotetion. It is the usual ratie that while seleting a samling insetion lan, to fix the oerating harateristi (OC) urve in aordane with the desired degree of disrimination. The samling lan is in turn fixed through suitably hosen arameters. The entry arameters used in the aetane samling literature are aetable quality level (AQL), indifferene quality level (IQL), limiting quality level (LQL) and maximum allowable erent defetive (MAPD), Maximum allowable average outgoing quality (MAAOQ), Average outgoing quality (AOQL). Several authors have rovided roedures to design the samling lans indexed through these arameters for various aetane samling lans. The onet of MAPD ( ) was introdued by Mayer (967) and further studied by Soundararajan (975) is the quality level orresonding to the infletion oint on the OC urve. The degree of sharness of insetion about this quality level is measured through t, the oint at whih the tangent to the OC urve at the infletion oint uts the roortion defetive axis. One of the desired roerties of an OC urve is that the derease of should be slower for lesser P a values of and faster for greater values of. If we set as the quality standard, the above roerty of the OC urve is obtained sine orresonds to the infletion oint of the OC urve and hene d Pa / d = 0 for = d Pa / d < 0 for < d Pa / d > 0 for > The mixed samling lan has been designed under two ases of signifiant interest. In the first ase samle size n is fixed and a oint on the OC urve is given. In the seond ase lans are designed when two oints on the OC urve are given. The roedure for designing the mixed samling lans to satisfy the above mentioned onditions was rovided by Shilling (967). Using Shilling s roedure, Devaarul (003) has onstruted tables for mixed samling lans (indeendent ase) having various samling lans as attribute lans. Radhakrishnan and Samath Kumar (006a, 006b, 007, 009) and Radhakrishnan et al., (00) have made ontributions to mixed samling lan for indeendent ase. Samath Kumar (007) has onstruted mixed samling lan with various samling lans as attribute lans using Poisson distribution as a base line distribution. The weighted Poisson distribution lays an imortant role in the aetane samling, mainly in the onstrution of samling lans. Eah outome (number of defetives) is seifi but an be assigned with different weights based on its imortane or usage. In using weighted Poisson 07 ISSN (Online): 49-64X (Print) - Rising Researh Journal Publiation

2 distribution with weights X α, α = the range of the distribution urtailed to,, 3 from 0,, This distribution an be viewed as a trunated Poisson distribution trunated at x =0. It will be more useful to the industries whih onentrates on seond s quality lots and also to the industries whih has atleast one defetive in the majority of the manufaturing lots. Even though the modern tehnologies aim at zero defetive/ defets but ratially it is very diffiult to make the lot as zero defetive lot. In this ontext, the aliation of weighted Poisson distribution in the onstrution of samling lans is very relevant and it has many features /advantages also. Radhakrishna Rao (977) suggested a weighted Binomial distribution an be used in designing samling lans. Sudeswari (00) studied the onstrution of samling lans using weighted Poisson distribution as a base line distribution. Mohana Priya (008), Radhakrishnan and Mohana Priya (008 a, 008 b) have onstruted the samling lans using weighted Poisson distribution as a base line distribution. In this aer, the mixed samling lan (indeendent ase) with double samling lan as attribute lan is onstruted using weighted Poisson distribution as a base line distribution. Tables are onstruted for the mixed samling lan indexed through i) MAPD, MAAOQ (ii) MAPD, AOQL. The lan indexed through (MAPD, MAAOQ) is also omared with the lan indexed through (MAPD, AOQL). GLOSSARY OF SYMBOLS The symbols used in this aer are as follows: : submitted quality of lot or roess Pa () : robability of aetane for given quality : maximum allowable erent defetive t : the oint at whih the infletion tangent of the OC urve uts the axis m : the rodut quality at whih average outgoing quality is maximum. h : relative sloe at n : samle size for the variable samling lan n, : first samle size for the attribute samling lan n, : seond samle size for the attribute samling lan : first attributes aetane number P a n r ( ) ( ) ( ) e n Pn x A P n x A r ( r )! CONSTRUCTION OF MIXED SAMPLING PLAN HAVING DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN USING WEIGHTED POISSON DISTRIBUTION The detailed roedure adoted in this aer for the onstrution of mixed samling lan having double samling as attribute lan using weighted Poisson distribution indexed through MAPD is given below: Global J. of Arts & Mgmt., 0: () : seond attributes aetane number β j : robability of aetane for lot quality j β j' : robability of aetane assigned to first stage for erent defetive j β j'' : robability of aetane assigned to seond stage for erent defetive j k : variable fator suh that a lot is aeted if X A = U - k OPERATING PROCEDURE OF MIXED SAMPLING PLAN HAVING DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN Shilling (967) has given the following roedure for the indeendent mixed samling lan with uer seifiation limit (U) and standard deviation (σ).. Determine the arameters of the mixed samling lan n, n,, n,, k,, and.. Take a random samle of size n from the lot. 3. If a samle average X A = U - k, aet the lot 4. If the samle average X > A = U - k, take a seond samle of size n, and ount the number of defetives d in the samle. 5. If the number of defetives d, aet the lot. 6. If the number of defetives d >, rejet the lot. 7. If + d, take a seond samle of size n, from the remaining lot and find the number of defetives d. 8. If d +d, aet the lot, otherwise rejet the lot. The OC funtion of the mixed samling lan, suggested by Shilling (967) for single samling lan is Pa Pn ( x A) Pn ( x A) P( j; n)..() 08 ISSN (Online): 49-64X (Print) - Rising Researh Journal Publiation j0 Equation () an be exressed as β j = β j'+ (- β j') β j''. where βj= P a, βj'= P n ( x A), - β j'= P n ( x A) and β j''= P j; n ) j0 ( By taking the double samling lan as attribute lan, equation () an be written as n k k n r..() ( )! ( )! k k r r Assume that the mixed lan is indeendent Deide the samle size n (for variable samling lan) to be used. Calulate the aetane limit for the variable samling lan as A U [ z( ' ) { z( ) / n }], where z is standard normal variate orresonding to t

3 Global J. of Arts & Mgmt., 0: () suh that t = z( t ) e u du Slit the robability of aetane β as β ' and β ''suh that β = β ' + (- β ') β ''. Fix the value of β ' Determine β '', the robability of aetane assigned to the attribute lan assoiated with the seond stage samle as β ''= (β - β ')/ (- β'). Determine the aroriate seond stage samle of n from the relation n r n k k n r β '' = r ( r)! k ( k )! r ( r )! Using the above roedure, tables have been onstruted to failitate easy seletion of mixed samling lan using double samling lan as attribute lans indexed through MAPD. Constrution of Tables The OC funtion of weighted Poisson distribution for single samling lan is given by, x ( x, ) Pa ; x=0,,.(3) x ( x; ) x0 The robability of aetane for double samling lan under weighted Poisson distribution when α = is used in this aer for determining the seond stage robabilities and is given by Pa () = n r ( ) n k k n r e n. β j''. (4) r ( r)! k ( )! ( )! k r r Using the equations (4) for n,= n,= n (say) the infletion oint ( ) is obtained by using 3 d a d a 0 and 0. 3 d d This setion rovides the roedure of onstruting the mixed samling lans having double samling lan as attribute lan indexed through (MAPD, MAAOQ) and (MAPD, AOQL). The average outgoing quality (AOQ) funtion is given by AOQ () =. Pa () MAAOQ = Pa () and AOQL = m Pa (m) The values of β '', n, nmaaoq, naoql are alulated for different ossible ombinations of, and n for β ' = 0.35 using C rogram and resented in Table. Seletion of the Plan Table is used to onstrut the lans for a seified MAAOQ and MAPD ( ). For any given values of and MAAOQ one an find the ratio R =MAAOQ/. Using Table, selet the nearest value of R and the orresonding values of and values are noted. The n value is determined using n=n/. Table is used to onstrut the lans for a seified AOQL and MAPD ( ). For any given values of and AOQL one an find the ratio R =AOQL/. Using Table, selet the nearest value of R, the orresonding and values are noted and n value is determined using n= n/. Examle : Given =0.04, MAAOQ = 0.0 and β ' = Find the ratio R = MAAOQ/ = 0.3 and selet the nearest value of R using Table as whih is assoiated with =, =6. From this one an find n= n / =3.667/0.04 =79.Thus n, =79, n, =79, =, =6 are the arameters seleted for the mixed samling lan with onditional double samling lan as attribute lan for a seified =0.04, MAAOQ = 0.0. Examle : Given =0.0, AOQL = 0.00 and β ' = Find the ratio R =AOQL/ = 0.5 and selet the nearest value of R using Table as 0.50 whih is assoiated with = 5, =8. From this one an find n= n/=5.049/0.0=5.thus n,=5, n,=5, =5, =8 are the arameters seleted for the mixed samling lan with double samling lan as attribute lan for a seified =0.0, AOQL = COMPARISON OF MIXED SAMPLING PLAN WITH DOUBLE SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROUGH (MAPD, MAAOQ) AND (MAPD, AOQL) In this setion mixed samling lan with double samling lan as attribute lan indexed through (MAPD, MAAOQ) is omared with the mixed samling lan with double samling lan as attribute lan indexed through (MAPD, AOQL) by fixing the arameters and MAAOQ = AOQL. Examle 3: Given MAPD = 0.04, MAAOQ = 0.06= AOQL and β ' = Find the ratio R = R = 0.4 and selet the nearest values of R and R orresonding to =, from Table as R = and R = whih are assoiated with =, =3 and =, =5 resetively. From this one an find the seond stage samle size n=n / =.8769/0.04 = 47for =, =3 and =0.04 and n = n / =.579/0.04 = 64 for =, =5 and = Thus n,= 47, n,= 47 and =, =3 are the arameters seleted for the mixed samling lan with double samling lan as attribute lan for a seified MAPD = 0.04, MAAOQ= 0.06 and n, =64, n, =64 and =, = 5 are the arameters seleted for the mixed samling lan with double samling lan as attribute lan for a seified MAPD = 0.04, AOQL=0.06. CONSTRUCTION OF AOQ CURVE The AOQ urves for the lans n=47, =, =3 (indexed through MAPD, MAAOQ) and n=64, =, =5 (indexed through MAPD, AOQL) based on the different values of n and AOQ are resented in Figure. 09 ISSN (Online): 49-64X (Print) - Rising Researh Journal Publiation

4 Global J. of Arts & Mgmt., 0: () CONCLUSION In this aer the roedure for onstruting mixed samling lans with double samling lan as attribute lan indexed through (MAPD, MAAOQ) and (MAPD and AOQL) with weighted Poisson distribution as the baseline distribution are resented and omared. Suitable tables are also rovided for easy seletion of the lans for the engineers who are working on the floor of the assembly. It is onluded from the study that the seond samle size required for mixed samling lan with double samling lan as attribute lan indexed through (MAPD,MAAOQ) is less than that of the seond stage samle size of the mixed samling lan with double samling lan as attribute lan indexed through (MAPD,AOQL), justified by Samath Kumar (008). These lans definitely hel the roduers, beause of the lesser samle size whih diretly result in lesser samling ost and indiretly redues the total ost of the rodut. REFERENCES Devaarul, S Certain studies relating to mixed samling lans and reliability based Samling lans. Ph.D., Dissertation, Deartment Of Statistis, Bharathiar University, Coimbatore, Tamil Nadu, India. Mayer, P.L A note on sum of Poisson robabilities and an aliation, Industrial Quality Control, 9 (5):-5. Mohana Priya, L Constrution and Seletion of aetane samling lans based on weighted Poisson distribution. Ph.D., Dissertation, Deartment of Statistis, Bharathiar University, Coimbatore, Tamil Nadu, India. Radhakrishna Rao, C A natural examle of a Weighted Binomial distribution, The Amerian Statistiian, 3 (): -5. Radhakrishnan, R. and Mohana Priya, L. 008 a. Seletion of single samling lan using weighted Poisson distribution, International Journal of Statistis and Systems, 3 (): Radhakrishnan, R and Mohana Priya, L. 008 b. Comarison of CRGS lans using Poisson and weighted Poisson distribution, Probstat Forum, ():50-6. Radhakrishnan, R. and Samath Kumar, R. 006 a. Constrution of mixed samling lan indexed through MAPD and IQL with single samling lan as attribute lan, National Journal of Tehnology, ():6-9. Radhakrishnan, R. and Samath Kumar, R. 006 b. Constrution of mixed samling lans indexed through MAPD and AQL with hain samling lan as attribute lan, STARS Int. Journal, 7():4-. Radhakrishnan, R. and Samath Kumar, R Constrution of mixed samling lans indexed through MAPD and IQL with double samling lan as attribute lan, The Journal of the Kerala Statistial Assoiation, 8:3-. Radhakrishnan, R. and Samath Kumar, R Constrution and omarison of mixed samling lans having ChSP-(0, ) Plan as Attribute lan, The International Journal of Statistis and Management System, 4(-): Radhakrishnan, R. Samath Kumar, R. and Malathi, M. 00. Seletion of mixed samling lan with TNT- (n, n ; 0) lan as attribute lan indexed through MAPD and MAAOQ, International Journal of Statistis and system, 5(4): Samath Kumar, R Constrution and Seletion of mixed variables-attributes samling lans. Ph.D., Dissertation, Deartment of Statistis, Bharathiar University, Coimbatore, Tamil Nadu, India. Shilling, E.G.967. A general method for determining the oerating harateristis of mixed variables-attributes samling lans single side seifiations, S.D.Known. Ph.D., Dissertation- Rutgers- The State University, New Brunswik, New Jersy. Soundararajan, V Maximum allowable erent defetive (MAPD) single samling insetion by attribute lan, Journal of Quality Tehnology, 7(4):73-8. Sudeswari, 00. Designing of Samling Plan Using Weighted Poisson distribution, M.Phil. Dissertation, Deartment of Statistis, PSG College of Arts and Siene, Coimbatore, Tamilnadu, India. 0 ISSN (Online): 49-64X (Print) - Rising Researh Journal Publiation

5 Table. Certain Parametri Values of Mixed Samling Plan for β ' =0.35 Global J. of Arts & Mgmt., 0: () β'' n nmaaoq naoql R R ISSN (Online): 49-64X (Print) - Rising Researh Journal Publiation