K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. Desig of Miimum Average Total Ispectio Samplig Plas K. Govidaraju, Istitute of Iformatio Scieces ad Techology Massey Uiversity New Zealad ABSTRACT The desig of a samplig ispectio pla that ivolves the miimum average total ispectio at a give quality level is of practical iterest. The acceptace samplig literature suggests that a specific type of double samplig pla be used for this purpose. This double samplig pla allows o ocoformig uits i the first stage ad o more tha oe ocoformig uit i the secod stage of ispectio. I this ote it is argued that the zero acceptace umber sigle samplig pla is still preferred to a double samplig pla for achievig miimum average total ispectio at a give poit o the Operatig Characteristic curve. A theoretical discussio o the broke sample size is provided for justifyig the choice of the sigle samplig pla. Key Words: Attribute ispectio, Average total ispectio, OC fuctio, Sigle ad double samplig plas, Zero acceptace umber.. INTRODUCTION Zero acceptace umber sigle samplig plas are largely used i practice due to several reasos (see Schillig []). These plas tolerate o ocoformity i the sample ad are psychologically appealig to the cosumer. A zero acceptace umber pla is very simple to use whe compared to the double, multiple ad sequetial samplig plas. For ay give poit o the Operatig Characteristic (OC) curve, usually for a give Limitig Quality Level (LQL) ad cosumer s risk β, zero acceptace umber plas require the miimum sample size. For a discussio of this miimum sample size property, see Hah [] ad Dodge [3]. A zero acceptace umber pla has also disadvatages such as lack of discrimiatio betwee good ad bad quality batches. Hece a zero acceptace umber pla is more useful for cosumer protectio whe the icomig quality is maitaied at good levels (see Schillig [] for a discussio). Oe of the popular sources for the desig of zero acceptace umber plas is Squeglia [4]. Zero acceptace umber plas are ot ofte recommeded i the literature wheever the o-accepted lots are screeed or rectified. This is because zero Correspodece: K. Govidaraju (Raj), SST 3.30, Istitute of Iformatio Scieces ad Techology (IIST), Massey Uiversity, Private Bag, Palmersto North, New Zealad; Email: k.govidaraju@massey.ac.z.
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. acceptace umber plas may ot ivolve the miimum Average Total Ispectio (ATI) for certai desig requiremets. That is, eve though the sample size is a miimum for the zero acceptace umber pla, the amout of ispectio eeded for screeig or rectifyig the rejected batch may be large leadig to a higher overall or total ispectio effort. For a expository discussio o this issue, see Baker [5]. Craig [6] addressed a specific issue of desigig a samplig pla whe miimum ATI at a give poit o the OC curve is of iterest. It was argued i favor of a double samplig pla i which o ocoformig uits are allowed i the first stage, ad o more tha oe ocoformig uit is allowed i the secod (fial) stage of ispectio. I this ote, we will show that the zero acceptace umber pla actually achieves the miimum average total ispectio (ATI) at a give quality level ad is preferred to the double samplig pla.. MINIMUM ATI DESIGN I acceptace samplig, a Type A lot ispectio implies samplig a isolated lot (or a short series of lots) where o quality history is geerally available. Type A lots iclude itermittetly supplied lots, lots of differet date coded items, ad oe-off or job lots of product. O the other had, Type B lots are formed from a statistically stable productio process series as idetifiable lots. I a Type B situatio, the lots are themselves radom samples from the productio process whose true fractio ocoformig is expected to be a costat. For further discussio o Type A ad Type B lots, see Stephes [7, pp.05-06]. The miimum ATI desig is ofte eeded i a Type B situatio of ispectig a series of lots. I a Type A cotext, the cocept ATI is meaigful oly if we view it as a average for ifiite submissios of the same quality lot. Oe of the popular tables for miimum ATI ispectio is that of Dodge ad Romig [8]. These tables provide sigle ad double samplig plas where the ATI is miimized at the actual or computed process average fractio ocoformig level (say, p ). The tabulated plas are desiged to offer protectio to the cosumer by fixig the Lot Tolerace Percet Defective (LTPD) at which the probability of acceptace is oly 0%. Alteratively Dodge-Romig tables ca also be used for the desired Average Outgoig Limit (AOQL) (istead of LTPD). The AOQL protectio to the cosumer is a worse log ru average quality protectio, while the LTPD desig gives idividual lot quality protectio. For a quick review of Dodge-Romig tables, see Stephes (7, p.93-05). Craig s [6] desig approach is differet from that of Dodge ad Romig [8] i that the ATI is miimized at a prescribed quality level as agaist the process average fractio ocoformig. The chose poit o the OC curve ca represet a poor quality such as the LTPD that has to be largely rejected or it ca be simply the average icomig quality of the lots. As such Craig s desig approach is simpler but it does ot deal with the AOQL type of protectio to the producer or two poits o the OC curve. A discussio o the desig of sigle ad double samplig plas that achieve the miimum ATI at a give poit o the OC curve is preseted i the succeedig paragraphs. Followig the covetio of the Iteratioal Orgaizatio for Stadardizatio (ISO) stadards, let the parameters of the sigle ad double attributes samplig pla be deoted as follows:
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. = sample size of the sigle samplig pla Ac = Acceptace umber of the sigle samplig pla i = the sample size for the i th stage of samplig (i =, ) of the double samplig pla. Ac i = the acceptace umber for the i th stage of samplig (i =, ) of the double samplig pla. Re i = the rejectio umber for the i th stage of samplig (i =, ) of the double samplig pla. Note that Re = Ac +. For a give fractio ocoformig p of the lot(s), let P ( ) a p be the OC fuctio givig the probability of acceptace at p. The objective is to desig a samplig pla that achieves the miimum ATI at a prescribed quality level, say p = p. It is also ecessary to prescribe the desired Pa ( p ) at p = p, say L( p ), ad the lot size N (say) for the desig of the pla. Craig [6] preseted the theory for fixig the acceptace ad rejectio umbers as Ac =, ad Ac =, ad obtaied the sample sizes ad with the followig Steps: P is as ear as possible to ( ). Obtai the value of such that 0, L p without exceedig it, where P 0, is the probability of fidig o ocoformig uits (ocoformities) i a sample of size.. Compute the value of, by trial ad error, such that P0, + P, P 0, is closest to the desired L( p ) where P, is the probability of gettig exactly oe ocoformig uit i a sample of size ad P 0, is the probability of fidig o ocoformig uits i a sample of size. Craig [6] did ot provide ay closed form solutios for ad. However this ca be easily obtaied uder the Poisso ad biomial models for the OC curve. Let us first cosider the Poisso model, which is exact for both Type A situatio of isolated lots ad Type B situatio of series of lots whe the attributes ispectio results i the umber of ocoformities (defects) per uit (area of opportuity). p Uder the Poisso model for OC curve, we have P0, = e. From Step of Craig s [6] desig procedure give earlier, oe ca provide a closed form solutio for the first sample size as ( ) l L p = it +. () p where it{ x } meas the iteger part of x. Note that we added i Eq. () to esure l L( p ) that P does ot exceed L( p ). I the evet of beig itself a iteger 0, p without ay fractios, there is o eed to add ad the secod sample is also ot eeded. 3
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. p Give P, = pe p, ad P0, = e, the secod sample size ca be foud from the OC fuctio of the double samplig pla with Ac =, ad Ac = amely ( p) P P P P e pe ( + ) p p a = 0, +, 0, = +. () =, P ( p ) must be equal to the desired ( ) At p p is give by a p ( p ) e L p, ad hece the solutio for L l p =. (3) p Whe biomial distributio is relevat (that is, for fractio ocoformig i a series of lots) the formula sample sizes ad are L( p ) ( p ) l = it + l ad (4) L p l p = ( ) ( p) ( p ) l ( p ). (5) Hypergeometric distributio is appropriate whe the lot size is very small with o quality history (that is ispectio of a Type A lot). Whe the hypergeometric distributio is used, we eed to follow a search procedure usig software ad o closed form expressio for ad ca be obtaied. For example, if the lot size N is oly 50, ad for give p = 0.06 ad L( p ) = 0.9, the double samplig pla with = ad = 9 (with Ac =, ad Ac = ) is recommeded i Craig [6] to miimize the ATI at p = 0.06 (the achieved miimum ATI beig 7.3). 3. BROKEN SAMPLE SIZE AND OTHER DESIGN ISSUES It should also be oted that a ( ) lies withi the ( ) a P p for Craig s Ac =, ad Ac = pla P p of the zero acceptace umber sigle samplig plas with sample sizes ad. This is due to the costraied choice of. Note that has to be fixed i such a way that Pa ( p ) does ot exceed the desired L( p ) value. A decrease i the sample size leads to a icrease i the probability of acceptace. P p < P p < P p is always true. Sice the ATI of Hece the iequality ( ) ( ) ( ) 0, a 0, 4
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. the zero acceptace umber sigle samplig pla with sample size is give by the formula (see Stephes [7, p. 97]) ( P ) ATI = P + N, (6) 0, 0, the ATI of the Ac =, ad Ac = pla will also be bouded by the ATIs of the zero acceptace umber sigle samplig plas with sample sizes ad. For Sectio example, we have P 0, ( 0.06) = 0.9400, ad P 0, ( 0.06) = 0.884. The ATIs of the sigle samplig plas with sample sizes ad are foud to be 7.59 ad 3.94 respectively. Note that the sigle samplig pla with sample size provides the miimum ATI L p of 0.90. What the Ac =, ad but fails to exactly meet the desired ( ) Ac = pla really does is that it icreases the acceptace probability slightly usig the secod stage of samplig helpig to achieve the desired probability of acceptace at p. But this is ot of ay real practical advatage if we take the ature of the poit prescribed poit o the OC curve ito accout. If p represets a good quality, the we will prefer the probability of acceptace at p to be equal to or slightly higher tha the desired value. I the above example, if p = 0.06 represets acceptable quality levels, the we would like to prescribe a miimum acceptace probability of 0.9 rather tha desirig it to be exactly 0.9. Hece the zero acceptace umber pla with sample size is the preferred choice because it esures that P ( p ) L( p ) a >. If the prescribed poit represets a limitig (poor) quality, the oe would like the prescribed (low) acceptace probability ot to be exceeded. I such a case, the zero P p tha the desired acceptace umber pla that achieves a slightly smaller a ( ) L( p ) is preferable. The sample size for such a zero acceptace umber pla will be. Note that the approach of takig the ature of the prescribed poit o the OC curve ito the desig ad providig a slightly better (discrimiatig) pla is i vogue for log i the literature (see Guether [9]). Hamaker [0] provided a theoretical treatmet for broke sample sizes. His procedure ivolves a radomizatio for the sample size (as i the case of a radomized test i the statistical hypothesis testig literature). A samplig pla havig a broke sample size betwee ad, will be defied as follows. A probability ϕ for choosig the pla with sample size ad the complimetary probability of ϕ for choosig the pla with sample size will be prescribed. This quatity ϕ is foud i such a way that the desired L( p ) ca be exactly met. I the above example, ϕ ca be obtaied solvig the equatio ( ) ( ) L p = 0.9 = 0.9400ϕ + 0.884 ϕ (7) 5
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. asϕ = 0.3. That is, 30% of the time, we will be usig the pla with sample size, ad 70% of the time, we will be usig the pla with sample size. For this zero acceptace umber pla with a broke sample size of.7, the ATI achieved is obtaied usig the ATI of the sigle samplig plas with sample sizes ad (which are 3.94 ad 7.59 respectively) as 3.94 0.3+7.59 0.7 = 6.5. This is smaller tha the ATI of 7.3 achieved by the Ac =, ad Ac = double samplig pla. Hece we establish theoretically that oly a zero acceptace umber sigle samplig pla ca achieve the miimum ATI for a give poit o the OC curve. Double ad multiple samplig plas are useful oly for a two-poit desig where the first prescribed poit is o the top of the OC curve (protectig the producer) ad the secod poit is at the bottom of the OC curve (protectig the cosumer). For a give sigle poit o the OC curve, zero acceptace umber plas aloe provide the miimum sample size as well as achieve the miimum ATI. Ay icrease i the acceptace umber, eve fractioally, will result i more samplig (ad hece the overall ispectio) effort. This fact ca be verified usig the fractioal acceptace umber sigle samplig pla discussed i Govidaraju []. As show by Govidaraju [], the OC fuctio of the fractioal acceptace umber sigle samplig pla ( Ac,, ) 0 Ac ad the double samplig pla (, ; Ac, Re, Ac ) are the same whe = +, Re = Ac = ( Re ) = leavig Ac udefied (i.e., o acceptace is allowed i the first stage of samplig). The fractioal acceptace umber sigle samplig pla with sample size will have a acceptace umber equal to /. For Craig s example coditios, the fractioal 6 acceptace umber pla ( = 8, Ac = 7 ) achieves a probability of acceptace of 0.906 at p = 0.06. For this pla, the ATI is 7+(50-7) (-0.906) =.04, which is obviously ot the miimum. The double samplig pla Ac =, ad Ac = used by Craig also fractioally allows oe ocoformig uit ad hece it calls for more ispectio tha a zero acceptace umber sigle samplig pla. 4. SOME SIMULATION RESULTS Cosiderig a typical small lot of size 50 havig 3 ocoformig uits ( p = 0.06 ), a simulatio study was made applyig the double samplig pla with = ad = 9 (with Ac =, ad Ac = ). Table summarizes the results. The simulatio study achieved a ATI of 7.40 which compares well with the theoretical ATI of 7.3. Table provides the summary for the simulatio of the equivalet broke sample size zero acceptace umber pla ( =.7, Ac = 0) cosidered i the earlier sectio. The simulatio study achieved a ATI of 6.369 which compares well with the theoretical ATI of 6.5. 6
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. Table. Double samplig pla simulatio results (0,000 simulatios) Decisio Cout No. of uits No. of uits sample ispected totally ispected Accept (First Stage) 8797 Accept (Secod Stage) 96 3 3 Reject (First Stage) 3 50 Reject (Secod Stage) 984 3 50 total = 0000 weighted mea weighted mea = 5.4 = 7.40 (ATI) Stadard deviatio = 9.356 Stadard deviatio = 4.80 Table. Broke sample size pla simulatio results (0,000 simulatios) Decisio Cout No. of uits No. of uits sample ispected totally ispected ( =, Ac = 0) pla weighted mea = 3083 3.686 (ATI) Accept 94 Reject 69 50 ( =, Ac = 0) Pla weighted mea = 697 7.565 (ATI) Accept 65 Reject 80 50 total = 0000 weighted mea weighted mea =.69 = 6.369 (ATI) Stadard deviatio = 0.46 A compariso of Tables ad summaries cofirm the followig: Stadard deviatio = 4.36 (i) (ii) A zero acceptace umber pla ivolves the miimum ATI for a give poit o the OC curve whe compared to the double samplig pla. I Type B situatios, oe may use a radomizatio device ad adopt broke sample =, Ac = 0 pla was size zero acceptace umber plas. (I Table, the ( ) applied 30.83% of the simulatios ad the (, Ac 0) the 69.7% of the time) = = pla was applied While the variability i the overall ispectio effort is about the same for the broke sample size pla ad the double samplig pla, the sigle samplig strategy ivolves a much smaller variability i the samplig ispectio stage. This fact is clear whe we compare Colum 3 stadard deviatio values. After all, it is a well kow fact that the double samplig plas are more difficult to admiister whe compared to sigle samplig plas. 7
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. (iii) The prescribed poit represets somewhat a good quality level at which the probability of acceptace is preferred to be high. Hece it is desirable to roud the sample size dow, which further reduces the ATI. I Table, the pla ( =, Ac = 0) achieved a ATI of 3.686, which is less tha the ATI of the double samplig pla as well as the broke sample size pla. 5. AN EXAMPLE A zero acceptace umber pla desiged to achieve the miimum ATI is more likely to useful for small lot sizes. Lot sizig is more of a ecoomic ad admiistrative issue tha a statistical issue. Just-i-time (JIT) or flexible productio models ofte suggest use of small lot sizes. Small lot sizes are also somewhat idustry specific. Lot size sesitive or specific applicatios of quality cotrol methods were reported by Burrows ad Silber [], Taylor [3], Taub [4], Schillig [], Combs ad Stephes [5], ad Foster [6]. The use of zero acceptace umber pla is particularly suitable for achievig miimum ATI i all lot sesitive applicatios metioed above. The zero acceptace umber plas also achieve the miimum ATI for large lot sizes as log as the desired protectio is i terms of a sigle specified quality level such the AQL or LTPD. Klaasse [7] described a acceptace samplig cotext where idepedet laboratories hallmark jewellery items submitted small lots. A zero acceptace umber pla was used maily to provide cosumer protectio. Klaasse s [7] credit based approach ad other work o the cotiuous samplig pla are ot of iterest to this paper but oly the cotext. For the isolated lot quality protectio, a zero acceptace umber pla with a sample of size = 45 is listed i Table of Klaasse [7] for a lot of size 00. The desired cosumer protectio was that a batch cotaiig 5% ocoformig be rejected 95% of the time. Usig the otatios used i this paper, the stipulated coditio meas p = 0.05 ad L( p ) = 0.05. Followig the approach give i sectio of this paper, we obtai the first sample size as = 45. That is, is the miimum sample size for which the hypergeometric probability P 0, 5 95 0 00 = 0.05 (8) The secod sample size has to be obtaied solvig 5 95 5 95 4 50 0 45 45 0 P0, + P, P 0, = + = 0.05 00 00 50 45 45 (9) or 8
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. 4 50 0 0.05 0.046063 50 0.0385 = = 0.0860. (0) Eq. (0) provides = 3. The desiged double samplig pla with = 45, = 3, Ac =, ad Ac = achieves a ATI of 93.4 (usig biomial distributio). If oe uses the zero acceptace umber sigle samplig pla with = 45, the ATI is slightly lower at 9.3. This pla also has a smaller probability of acceptace of 0.046 at the udesirable lot quality of 5% ocoformig. Hece we cofirm agai that the use of the double samplig approach is uecessary whe a sigle poit ( p, Pa ( p ) ) o the OC curve is prescribed. A o-zero acceptace umber is required oly whe the desired protectio ivolves more tha oe poit o the OC curve or coditios such as the AOQL are prescribed. For a discussio ad examples o such type of miimum ATI desigs, see Wu et. al., [8], Guether [9], Duca [0, pp. 308-30), ad Dodge ad Romig [8, pp.7-0 ad 33-43]. 6. CONCLUSION This ote establishes o various grouds that the zero acceptace umber sigle samplig pla is the preferred choice if miimum ATI (or sample size) is of iterest for a prescribed sigle poit o the OC curve, which may either represet a good or bad quality. However, a zero acceptace umber plas are ot a preferred choice if two poits are prescribed or other ecoomic criteria are prescribed for the desig of the samplig pla. ACKNOWLEDGEMENT The author is thakful to the referees for commets, particularly the suggestio to perform a simulatio ad the example. REFERENCES [] Schillig E.G. A lot sesitive samplig pla for compliace testig ad acceptace ispectio. Joural of Quality Techology, 978, 0 (), 47-5. [] Hah, G. J. Miimum size samplig plas. Joural of Quality Techology. 974, 6 (3), -7. [3] Dodge, H. F. Chai samplig ispectio pla. Idustrial Quality Cotrol. 955, (4), 0-3. [4] Squeglia, N. L. Zero Acceptace Number Samplig Plas, ASQ Quality Press, America Society for Quality, Milwaukee, Wiscosi, 994. [5] Baker, R. C. Zero acceptace samplig plas: expected cost icreases. Quality Progress, 988, (), 43-46. 9
K. Govidaraju (005). Desig of Miimum Average Total Ispectio Samplig Plas. Commuicatios i Statistics: Simulatio ad Computatio, 34(), 485-493. [6] Craig, C. C. A ote o the costructio of double samplig plas. Joural of Quality Techology, 98, 3 (3), 9-94. [7] Stephes K. S. The Hadbook of Applied Acceptace Samplig- Plas, Priciples, ad Procedures, ASQ Quality Press, America Society for Quality, Milwaukee, Wiscosi, 00. [8] Dodge, H. F.; Romig, H. G. Samplig Ispectio Tables, Sigle ad Double samplig. d ed., Joh Wiley ad Sos, New York, 959. [9] Guether, W. C. Use of the biomial, hypergeometric ad Poisso tables to obtai samplig plas. Joural of Quality Techology, 969, (), 05-09. [0] Hamaker, H. C. The theory of samplig ispectio plas. Philips Techical Review, 950, (9), 60 70. [] Govidaraju, K. Fractioal acceptace umber sigle samplig pla. Commuicatios i Statistics Simulatio ad Computatio, 99, 0 (), 73 90. [] Burrows, G. L.; Silber C. Tolerace limits for small lots. Idustrial Quality Cotrol, 963, 9 (8), 6-0. [3] Taylor, E. F. A special case of percetage samplig. Idustrial Quality Cotrol, 964, 0 (0), 3-4. [4] Taub, T. W. Battery ispectio by variables. Joural of Quality Techology, 976, 8 (), 74-80. [5] Combs, C. A. ad Stephes L. J. Upper Bayesia cofidece limits o the proportio defective. Joural of Quality Techology, 980, (4), 96-00. [6] Foster, G. K. Implemetig SPC i low volume maufacturig. ASQC Aual Quality Cogress Trasactios, 988, 4, 6-67. [7] Klaasse, C. A. J. Credit i acceptace samplig o attributes. Techometrics, 00, 43 (), -. [8] Wu, Z.; Xie, M.; Wag, Z. Optimum rectifyig ispectio plas. Iteratioal Joural of Productio Research, 00, 39 (8), 575-588. [9] Guether, W. C. Determiatio of rectifyig ispectio plas for sigle samplig by attributes. Joural of Quality Techology, 984, 6 (), 56-63. [0] Duca, A.J. Quality Cotrol ad Idustrial Statistics, 3 rd Ed., Irwi, Illiois, 965. 0