Estimation of apparent fraction defective: A mathematical approach

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1 Availabl onlin at Plagia Rsarch Library Advancs in Applid Scinc Rsarch, 011, (): ISSN: CODEN (USA): AASRFC Estimation of apparnt fraction dfctiv: A mathmatical approach 1 D.V.Ramana, M.V.Ramanaiah, 3 S.K. Khadar Babu *, 4 K.Karthikyan, 5 B.Rajsh Anand 1,,5 S.V.Univrsity, Tirupathi, India 3,4 VIT Univrsity, Vllor, India ABSTRACT In this articl, an attmpt was mad to stimat th apparnt fraction dfctiv whn th inspction risks ar unknown using Bta distribution of first kind truncatd at point b. This study uss to idntify th prformanc of sampling plans lik singl sampling plan and Doubl sampling plan. Ky words: Bta distribution, Truncation, Accptanc Quality Lvl, Gamma distribution. INTRODUCTION Accptanc Inspction is a part of Quality Assuranc through which Product Control is xrcisd. This is in contrast with Procss Control in which Control charts play th dominant rol to nsur a stat of statistical control of th procss. Accptanc Inspction is a ncssary part of a manufacturing systm and may b applid to incoming matrials, final products and to th smi itms in a production lin. Th word sampling inspction is usd whn th quality of th product is valuatd by sampling rathr than 100% inspction ( Gunthr W.C [1977]). Sampling plans uss a random sampl as th basis for assssing th quality of a finit population of units calld a lot(h.f.dodg[1943]). Th supplir of th lot is gnrally calld th producr and th buyr is calld th consumr. Accptanc Sampling is a statistical procdur that spcifis a rul to accpt or rjct a lot, basd on th quality obsrvd in th sampl drawn from that lot. That is why it is calld lot sntncing procdur. A sampling plan is thus a st of ruls to xcut accptanc inspction. Svral basics idas of Accptanc sampling can b found in Montgomry (1997). Plagia Rsarch Library 84

2 S. K.Khadar Babu t al Adv. Appl. Sci. Rs., 011, ():84-89 PRELIMINERIES If th dcision about accpting or rjcting a lot is takn on th basis of only on sampl drawn from th lot, it is calld a singl sampling plan. This plan is basd on th following tchnical trms. a. Accptabl Quality Lvl (AQL) : This is th proportion of dfctivs with which a lot can b accptd. It is basd on th obsrvation that in spit of all th fforts mad to avoid non conformitis, crtain dfctivs occur in th lots and th consumr also agrs to accpt such lots. It is usually xprssd as a prcnt lik 1 % or 0.5% dfctivs bing admittd. It is convntionally dnotd by p 1. b. Rjctabl Quality Lvl (RQL) or Lot Tolranc Prcnt Dfctiv (LTPD) : This is th worst cas fraction dfctiv at which th consumr can accpt th lot. If th obsrvd fraction dfctiv touchr LTPD th lot is rjctd. It is dnotd by p and taks valus highr than AQL. C. Producr s Risk : Sinc th dcision on th lot is basd on random sampl, thr is vry possibility that on sampl may show a highr numbr dfctivs than anothr sampl drawn from th sam lot. Th Producr, aftr inspction may rjct a lot vn though th lot rally dos not warrant rjction! This is calld Typ-1 rror and th probability of committing such an rror is known as producr s risk. This is dnotd by α and givn by th conditional probability P( X c / P AQL). d. Consumr s Risk : It is th probability of accpting a lot, basd on sampl, givn that th lot truly contains LTPD. This rror, known as Typ-II rror, occurs bcausth sampl might som tims fail to rflct th ral quality of th lot. Th risk of committing this rror is known as consumr s risk, dnotd by β and givn by th conditional probability P( X c / P LTPD). NEW APPROACH Whil dfining th apparnt fraction dfctiv, it is assumd that th typ-i and typ-ii inspction risks ar known and fixd(sk.khadar Babu 007). In practic, variation in th valus of ϕ and ε occur du to svral uncontrollabl factors. Whn th inspctor changs th gag or inspctor is changd from th tsting station or th oprating nvironmnt gts disturbd, it is possibl that th inspction risks ar draggd to on of th xtrms say 0 or 1. In othr words ϕ and ε my com closr 0 or 1. It is thrfor, rasonabl to dscrib th inspction risk as a continuous random variabl Y, 0 Y 1. W can us ithr Uniform distribution in [0,1] or Bta distribution of typ 1 to dscrib that bhavior of Y. In th following sction w us typ 1 Bta distribution and xamin its proprtis to dscrib th uncrtainty in ϕ and ε. Plagia Rsarch Library 85

3 S. K.Khadar Babu t al Adv. Appl. Sci. Rs., 011, (): Bta Distribution and its proprtis A continuous random variabl Y is said to hav a Bta distribution of typ 1 with paramtr (m, if its probability dnsity function (pdf) is givn by 1 m 1 n 1 f y (1 y),0 y 1 β ( m, 1 m 1 n 1 Whr β ( m, y (1 y) dx. Th valu of β ( m,, for m,n positiv intgrs is givn by β ( m, 0+ m). m + Sinc n ) ( n 1)!, w gt th rlationship ( m 1)!( n 1)! β ( m, ( m + n 1)! Th distribution function of Y is givn by G(y) p(y< y) 1 G( y) β ( m, y 0+ y m 1 (1 y) n 1, dx,0 y 1 Th r th momnt of Y about origin o can b shown to b qual to E r + m). m + ( y r ) r + m +. Γ ( m) m Hnc E and m + n mn V ( m + ( m + n + 1) If Y Follows β ( m, thn(1 y) follows β ( n, m), Whn m n 1, thdistribution Y will b uniform on[0,1]. 3. Th distribution of β (,) to dscrib ϕ and ε On of th particulars cass of Bta distribution of first kind β (,) in gnral is givn by f 6y(1 Y), o y 1 For this distribution E(y) ½ and V(Y) 7/0. Clarly E(Y) > V(y) Plagia Rsarch Library 86

4 S. K.Khadar Babu t al Adv. Appl. Sci. Rs., 011, ():84-89 This distribution can b usd as a modl to xplain ϕ and ε. If w assum that ach on of th inspction risks follow β (,) distribution thn it follows that 1 7 E ( ϕ) and V ( ϕ) 0 and 1 E ( ε ) and V ( ε ) 7 0 Using ths valus th xpctd apparnt fraction dfctiv dnotd by p ( 1 E( ϕ)) + (1 p) E( ε ) E( ) Π is givn by 1 p 1 + (1 p) 1 1 This is an xpctd rsult in which th apparnt fractoon dfctiv is found to b indpndnt of 1 th incoming lot quality. E ( ε ) implis that inspctor is indiffrnt in classifying th itm as good or bad. Similar is th cas with E (ϕ ) and ths two xpctd risks crat th highst uncrtainty in dcision making Estimating ϕ using β (,) distribution truncatd at b With rgard to th inspction risks, it is rasonabl to assum that ithr of th risks of misclassification is not mor than 0.1 or 0.. Ths valus corrspond to 1% and % risks of misclassification. It is also possibl that du to fatigu or monotony of inspction of th misclassification risks, som tims happn to b on th highr sid, starting with minimum of 0.5 or 0.6. This is only a thortical possibility but a good systm xpcts both th risk to b vry small. a. Uppr truncatd β (,) distribution : W dnot this distribution by β (,) and th pdf is givn by f u { 1y(1 y), 0 < y < 0. 5 For this distribution 5 E u and V u b. Lowr truncatd β (,) distribution For th lowr truncatd β (,) distribution w gt Plagia Rsarch Library 87

5 S. K.Khadar Babu t al Adv. Appl. Sci. Rs., 011, ():84-89 f L { 1y(1 y), 0.5 < y < 1 For this distribution V L In th following discussion w xamin this Bta(,) distribution and study thir ffct on th apparnt fraction dfctiv as wll as on th proprtis of th singl sampling paln. Whn both ϕ and ε ar truncatd on th uppr sid at b<1, th apparnt fraction dfctiv can b workd out in a closd form. Considr th following. p(6 1b + 6b ) + (4b 3b ) (6 4b) 3.4 Apparnt fraction dfctiv whn ϕ and ε ar at thir xpctd valus. In this sction w dtrmin E ( ) undr thr conditions. 1. ϕ Follows Uppr truncatd β (,,0.5) & ε Follows lowr truncatd β (,,0.5).. ϕ Follows lowr truncatd β (,,0.5) & ε Follows uppr truncatd β (,,0.5). 3. Both ar truncatd in on dirction ( Lowr) 4. Both ar truncatd in on dirction (Uppr) Substituting th valus of E(ϕ) and E(ε ) p( 1 E( ϕ)) + (1 p) E( ε ) and simplifying w gt th Following possibl valus. Tabl: Typs of inspction rrors Cas Typ of truncation for Valu of ϕ I Lowr Lowr II Lowr Uppr III Uppr Lowr IV Uppr Uppr ε p p Plagia Rsarch Library 88

6 S. K.Khadar Babu t al Adv. Appl. Sci. Rs., 011, ():84-89 CONCLUSION W obsrv th following rsults from th valu of for a. Whn ϕ and ε ar basd on th diffrnt typs of truncation th xpctd fraction dfctiv bcoms indpndnt of th incoming lot fraction dfctiv. ( Cas II and Cas III). b. whn th typ of truncation is changd from lowr to uppr th rsulting bcoms complmntary to th prvious combination. This is tru btwn Cas II and Cas III and also btwn Cas I and Cas- IV. REFERENCES [1] Dodg, H.F, Annals of Mathmatical Statistics, 1943, pp , 14. [] Gunthr, W.C., Sampling Inspction in Statistical Quality Control, Monograph, 1977, 37, Charls Griffin and Company Ltd. [3] Khadar Babu SK., On som Aspcts of Sampling Plans for attribut Inspction, Ph.D Thsis, S.V. Univrsity, Tirupathi, 007, pp [4] Montamry C. Douglas, Introduction to Statistical Quality Control, 3 rd Edition, John Wily & Sons, Inc, Nw York, 1997, pp Plagia Rsarch Library 89