International Journal of Engineering Research and Modern Education (IJERME) Impact Factor: 7.018, ISSN (Online): (

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1 CONSTRUCTION AND SELECTION OF CHAIN SAMPLING PLAN WITH ZERO INFLATED POISSON DISTRIBUTION A. Palansamy* & M. Latha** * Research Scholar, Department of Statstcs, Government Arts College, Udumalpet, Tamlnadu ** Prncpal (Red), Kamarajar Government Arts College, Suranda, Tamlnadu Cte Ths Artcle: A. Palansamy & M. Latha, Constructon and Selecton of Chan Samplng Plan wth Zero Inflated Posson Dstrbuton, Internatonal Journal of Engneerng Research and Modern Educaton, Volume 3, Issue, Page Number 46-50, 08. Copy Rght: IJERME, 08 (All Rghts Reserved). Ths s an Open Access Artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract: In ths artcle gven a constructon procedure of attrbute Zero Inflated Posson chan samplng plan- for varable fracton defectve s presented usng stochastc dfferental equatons. An teratve procedure for fndng the parameters of the plan satsfyng the gven condtons wth respect to producer qualty level s gven. Tables are constructed for easy selecton of the parameters whch are readly avalable to apply n the shaft floors of producton process. The performance of the chan samplng plan for varable fracton defectve s also dscussed by determnng the operatng characterstc functon. The procedure s developed to draw an Average Outgong Qualty Level (AOQL), curve by usng Zero Inflated Posson chan samplng plan- and compare the dfferent procedures n order to show the ambguty of the procedure and ther results. Key Words: Chan Samplng Plan, Zero Inflated Posson Dstrbuton & Average Outgong Qualty Level (AOQL). Introducton: The Modern developments n technology help to mprove the qualty of products and ther producton. Improvement n product qualty s also due to the well montorng of the producton process. Products not meetng the specfed qualty standards n these stuatons s a rare phenomenon. Samplng nspecton carred out n these case may result wth the nformaton consstng many zeros whch are the values of the number of nonconformng product. The Zero-Inflated Posson (ZIP) dstrbuton can be used as the approprate probablty dstrbuton to data consstng many over dspersed zeros. ZIP dstrbuton has been used n a wde range of dscplnes such as agrculture, epdemology, econometrcs, publc health, process control, medcne, manufacturng, etc. Some of the applcatons of ZIP dstrbuton can be found n Bohnng et al. (999), Dodge (955) has proposed Chan Samplng Plan n whch allows sgnfcant reducton n sample sze and the condton for a contnung successon of lots from a stable and trusted suppler. Lambert (99), and Yang et al. (0). Constructon of control charts usng ZIP dstrbuton are dscussed n Sm and Lm (008).Some theoretcal aspects of ZIP dstrbutons are mentoned n McLachlan and Peel (000). Sngle samplng plans by attrbutes under the condtons of Zero nflated Posson dstrbuton are determned by Loganathan and Shaln (03), Suresh and Latha (00) have gven a procedure and tables for the selecton of Bayesan chan samplng plan-. Soundararajan (978 a, b) has descrbed procedures and tables for constructon and Selecton of Chan samplng plans (ChSP-) ndexed by specfed parameters. Condtons for Applcaton of ChSP -: The cost of destructveness of testng s such that a relatvely small sample szes s necessary, although other factors make a large sample desrable. The product to be nspected comprses a seres of successve lots produced by a contnung process. Normally lots are expected t be of essentally the same qualty. The consumer has fath n the ntegrty of the producer. Operatng Procedure: The plan s mplemented n the followng way: For each lot, select a sample of n unts and test each unt for conformance to the specfed requrements. Accept the lot f d (the observed number of defectves) s zero n the sample of n unts, and reject f d>. Accept the lot f d s equal to and f no defectves are found n the mmedately precedng samples of sze n. Dodge (955) has gven the operatng characterstc functon of ChSP- as P a (p) = P 0 + P (P 0 ) Where Pj = probablty of fndng j nonconformng unts n a sample of n unts for j = 0.. The Chan samplng Plan s characterzed by the parameters n and. When =0, the OC functon of a ChSP - plan reduces to the OC functon of the Sngle Samplng Plan wth acceptance number zero and when = 0, the OC functon of ChSP- plan reduces to the OC functon of the Sngle Samplng Plan wth acceptance number. 46

2 Operatng Characterstc Functon of ZIP Model: The OC functon s defned as () Where p s the probablty of fracton defectve The numbers of defects are zero for many samples there may consder Zero nflated Posson probablty dstrbuton. The probablty mass functon of the ZIP ( dstrbuton s gven by Lambert (99) and McLachlam and peel (000) ) = f(x) + (- )P(X=x λ) () Where f(x) = and P( X x / ) =, when x =0,,, The probablty mass functon can also be expresses as ( ) e when x 0 P( X x, ) = x e ( ), when x,,...,0, 0 x! In ths dstrbuton, φ may be termed as the mxng proporton. φ and λ are the parameters of the ZIP dstrbuton. Accordng to McLachlan and Peel (000), a Zp dstrbuton s a specal knd of mxture dstrbuton. The OC functon of the condtons of ZIP ( dstrbuton can be defned as P c a ( p) ( ) e ( ) x x e, x 0, 0, 0. x! Where λ = np Chan Samplng Plans (ChSP-) wth Zero- Inflated Posson Dstrbuton: The probablty of acceptance for chan samplng plan of type ChSP- based on Zero- nflated Posson dstrbuton P ( p) ( ( ) e ) ( ( ) e ) ( ) e np ( ( ) e ) (4) a Average Outgong Qualty Lmt (AOQL): The AOQL of a samplng plan s maxmum value on the AOQ curve. It s applcable for defectve unts, defects per unt, and defects per quantty. It s expressed as ether a defectve rate (fracton defectve, percent defectve, dpm) or as a defect rate (defects per unt, defects per 00 unts, dpm). The AOQ curve gves the average outgong qualty (left axs) as a functon of the ncomng qualty (bottom axs). The AOQL s the maxmum or worst possble defectve or defect rate for the average outgong qualty. Regardless of the ncomng qualty, the defectve or defect rate gong to the customer should be no greater than the AOQL over an extended perod of tme. Indvdual lots mght be worst than the AOQL but over the long run, the qualty should not be worse than the AOQL. Thus Chan samplng plans Zero- nflated Posson dstrbuton Average Outgong Qualty (AOQ) s approxmated obtaned by AOQ = P ( p) naoq ( np np( ) e ) np( ( ) e ) ( ) e ( np) ( ( ) e ) (5) Dfferentatng AOQ wth respect to np and equatng to 0, the value of Average Outgong lmt (AOQL) can be obtaned by solvng the equaton. ( ) e (( ( ) e ) np ( ) e ( np ) ( ( ) e ) p a np ( np) e ( ) ( ( ) e )) np ( ) e = 0 (6) From Equaton (6) the values of np (=npm) can be calculated for dfferent values of and.substtutng npm n equaton (5) naoql values are obtaned. Comparson wth Posson Chan Samplng Plan (ChSP ): From Table for fxed value of φ,, the np value s must greater than Posson plan for hgher value of probablty of acceptance. When the probablty of acceptance s very low the np value for Zero Inflated Posson model s greater than but closer to that of Posson model. (3) 47

3 Regardng Average Outgong Qualty Lmt (AOQL) the small value of. The Average Outgong Qualty lmt (AOQL) s Zero Inflated Posson model s should be hgher than the Posson model as ncreases the value of Average Outgong Qualty Lmt (AOQL) s also ncreases, as value ncreases the Average Outgong Qualty Lmt (AOQL) for the Zero Inflated Posson model approaches to the Posson model. Table : Chan Samplng Plan under Zero - Inflated Posson P a (p) Table: Certan Parametrc values Chan Samplng Plan (ChSP-) wth ZIP Model np np np m naoql p / p AOQL/ p

4 Concluson: In a well montored process, most of the products wll meet the specfed qualty standards. Occurrence of Zero- nonconformng per product would be more frequent n ths samplng nspecton. A zero nflated model s the approprate probablty dstrbuton to the number of nonconformtes per product manufactured n such producton process. The Chan samplng plan gves more pressure on the producer f the qualty deterorates. These plans provde consumer an assurance regardng the outgong qualty or the qualty of the lot after the nspecton. Hence one can recommend ths type of samplng plans for better qualty control practce. Ths plan wll be more useful to the qualty control practoners to meet out the consumer requrements. References:. Bohnng, D., Detz, E., Schlattmann, P. (999). The zero-nflated Posson model and the decayed, mssng and flled teeth ndex n dental epdemology. Journal of Royal Statstcal Socety, Seres A 6: CLARK, C. R. (960), OC Curve for ChSP- Chan Samplng Plans, Industral Qualty Control, Vol.7, No. 4, pp Dodge H.F, Chan samplng nspecton plans, Industral Qualty Control II(4) (955),

5 4. Lambert, D. (99). Zero-nflated Posson regresson wth an applcaton to defects n manufacturng. Technometrcs 34: Loganathan, A and Shaln, K., (03). Determnaton of Sngle samplng plans by attrbutes under the condtons of Zero nflated Posson dstrbuton. Communcatons n Statstcs Smulaton and Computaton 43:3, Latha.M and Rajeswar M Cost and regret functon for Bayesan Chan Samplng Plan-, Internatonal Journal of Industral Engneerng and Technology Vol-4 No-3 pp Latha. M and S. Jeyabharath Selecton of Bayesan Chan Samplng Attrbutes Plans Based On Geometrc Dstrbuton, Internatonal Journal of Scentfc and Research Publcatons, Volume, Issue Soundararajan V Procedure and tables for constructon and selecton of chan samplng plan (ChSP-) part-, Journal of Qualty and Technology 0() (978) Soundararajan V Procedure and tables for constructon and selecton of chan samplng plan (ChSP-) part-, Journal of Qualty and Technology (978) Suresh K K and Latha M Constructon and evaluaton of performance measures for Bayesan chan samplng plan (BChSP-) Far East Journal of Theoretcal Statstcs (00) V. Sankara Subramanyan & K. Veerakumar, A Study on Rsk Management n Constructon Projects - An Emprcal Study, Internatonal Journal of Advanced Trends n Engneerng and Technology, Page Number -4, Volume, Issue, 07.. A. Palansamy and M. Latha,(08) Constructon of Bayesan Sngle Samplng Plan by Attrbutes under the Condtons of Gamma Zero Inflated Posson Dstrbuton, Internatonal Research Journal of Advanced Engneerng and Scence, Volume 3, Issue, pp. 67-7, 0 50