The OC Curve of Attribute Acceptance Plans


 Sybil Booker
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1 The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve Fgure 1 Typcal Operatng Characterstc (OC) Curve The Shape of the OC Curve The frst thng to notce about the OC curve n Fgure 1 s the shape; the curve s not a straght lne. Notce the roughly S shape. As the lot percent nonconformng ncreases, the probablty of acceptance decreases, just as you would epect. Hstorcally, acceptance samplng s part of the process between a part s producer and consumer. To help determne the qualty of a process (or lot) the producer or consumer can take a sample nstead of nspectng the full lot. Samplng reduces costs, because one needs to nspect or test fewer tems than lookng at the whole lot. Samplng s based on the dea that the lots come from a process that has a certan nonconformance rate (but there s another vew descrbed below). The concept s that the consumer wll accept all the producer s lots as long as the process percent nonconformng s below a prescrbed level. Ths produces the, so called, deal OC curve shown n Fgure 2. When the process percent nonconformng s below the prescrbed level, 4.0% n ths eample, the probablty of acceptance s 100%. For qualty worse than ths level, hgher than 4%, the probablty of acceptance mmedately drops to 0%. The dvdng lne between 100% and 0% acceptance s called the Acceptable Qualty Level (AQL). The OC Curve of Attrbute Acceptance Plans Page 1 of 7
2 The orgnal dea for samplng takes a smple random sample, of n unts, from the lot. If the number of nonconformng tems s below a prescrbed number, called the acceptance number and denoted c, we accept the lot. If the sample contans more nonconformng tems, we reject the lot % 4.0% 6.0% 8.0% % 14.0% 16.0% 18.0% Fgure 2 Ideal OC Curve The only way to realze the deal OC curve s 100% nspecton. Wth samplng, we can come close. In general, as the sample sze ncreases, keepng the acceptance number proportonal, the OC curve approaches the deal, as shown n Fgure 3. Smlarly, as the acceptance number, c, gets larger for a gven sample sze, n, the OC curve approaches the deal. Fgure 4 llustrates the relatonshp. Some Specfc Ponts on the OC Curve Because samplng doesn t allow the deal OC curve, we need to consder certan rsks. The frst rsk s that the consumer wll reject a lot that satsfes the establshed condtons,.e., the process qualty s acceptable, but, by the luck of the draw, there are too many nonconformng tems n the sample. Ths s called the producer s rsk, and s denoted by the Greek letter α. The second rsk s that the consumer wll accept a lot that doesn t meet the condtons,.e., by the luck of the draw there are not many nonconformng tems n the sample, so the lot s accepted. Ths s the consumer s rsk and s denoted by the Greek letter β. The lterature contans a varety of typcal values for α and β, but common values are 5% and 10%. When we locate these values on the OC curve, epressed n terms of probablty of acceptance, we actually locate 1 α. The OC Curve of Attrbute Acceptance Plans Page 2 of 7
3 n= 50, c=1 n=100, c=2 n=200, c=4 2.0% 4.0% 6.0% 8.0% % 14.0% Fgure 3 As n Increases the OC Curve Approaches the Ideal n=100, c=2 n=100, c=1 n=100, c=0 2.0% 4.0% 6.0% 8.0% % 14.0% Fgure 4 As c Increases, for Fed n, the OC Curve Approaches the Ideal The OC Curve of Attrbute Acceptance Plans Page 3 of 7
4 These ponts correspond to specfc values of lot qualty and they have a varety of names. The pont assocated wth 1 α s often called the Acceptable Qualty Lmt or AQL. Ths s not necessarly the same AQL used to descrbe the deal OC curve. For an α of 5% ths means a process operatng at the AQL wll have 95% of ts lots accepted by the samplng plan. Smlarly, the pont assocated wth β s often called, n contrast, the Rejectable Qualty Lmt or RQL. A process operatng at the RQL wll have 5% of ts lots accepted by the samplng plan. Lastly, some authors consder the process qualty where the lots have a 50% probablty of acceptance. Ths s called the Indfference Qualty Lmt or IQL. Fgure 5 llustrates these ponts α β AQL IQL RQL Fgure 5 Specfc Ponts on the OC Curve A Stream of Lots and the Bnomal Dstrbuton We descrbed the OC curve n terms of a process that produces a seres of lots. Ths leads us to recognze that the underlyng dstrbuton s the bnomal. In the bnomal dstrbuton, there are two possble outcomes. The tems n the sample are ether conformng or nonconformng. In addton, the probably of selectng a nonconformng tem doesn t change as a result of the sample. Snce we are samplng from a process, the potentally nfnte number of tems s not mpacted by takng the sample. When the producer presents lots for acceptance, they often come from a process that s operatng at some qualty level,.e., the process produces a certan percentage of nonconformng tems. The probablty of obtanng a specfed number of nonconformng tems, Pr(), from a sample of n tems wth percent nonconformng, denoted p, s gven by the bnomal dstrbuton. The OC Curve of Attrbute Acceptance Plans Page 4 of 7
5 n Pr( ) = p n ( 1 p), = 0,1,, n In a sngle sample plan we accept the lot f the number of nonconformng tems s c or less. Ths means we are nterestng n the probablty of 0, 1,, c tems. We wrte ths as Pr( c) = ( 1 p) The OC Curve of Attrbute Acceptance Plans Page 5 of 7 c = 0 n p The probablty of acceptng the lot s the probablty that there are c or fewer nonconformng tems n the sample. Ths s the equaton above, and s what we plot as the OC curve. The Isolated Lot And The Hypergeometrc Dstrbuton The bnomal dstrbuton apples when we consder lots comng from an ongong producton process. Sometmes we consder solated lots, or we are nterested n a specfc lot. In these cases, we need to realze that takng the sample, because we sample wthout replacement, changes the probablty of the net tem n the sample. In these cases, we need the hypergeometrc dstrbuton. Pr ( ) d N d n = N n Here, N s the lot sze, n s the sample sze, and d s the number of nonconformng tems n the lot. If we are nterested n determnng the probablty of c or fewer nonconformng tems n the sample then we wrte: c ( c) = = 0 d N d n N n We can use ths equaton to draw the OC curve for the solated lot. Pr Often, the β rsk s appled to each lot, nstead of the stream of lots. In these cases, the qualty level correspondng to a probablty of acceptance equal to β s called the Lot Tolerance Percent Defectve (LTPD). Sngle and Double Sample Plans The materal above dscusses samplng plans n whch we draw one sample from the lot. Ths s called a sngle sample plan. We descrbe the plan by a set of parameters: n s the sample sze,
6 c s the mamum number of nonconformng tems allowed for acceptance, and r s the mnmum number of nonconformng tems allowed for rejecton. In a sngle sample plan r and c dffer by 1. In contrast, there are double samplng plans n whch we take the frst sample and make one of three decsons: accept, reject, or take a second sample. If we take the second sample, we then make an accept/reject decson. As descrbed above the set of parameters used to descrbe a double sample plan are: n s the th sample sze, c s the mamum number of nonconformng tems allowed for acceptance on the th sample, and r s the mnmum number of nonconformng tems allowed for rejecton on the th sample. For eample, a sngle sample plan may be: A double sample plan may be: n = 20, c = 2, r = 3. n 1 = 20, c 1 = 1, r 1 = 4 n 2 = 20, c 2 = 4, r 2 = 5 In ths eample, f we had 2 nonconformng tems on the frst sample, we would draw the second sample. In total, we would have sampled 40 tems. We can calculate the probablty of acceptance, the nformaton we need to defne the OC curve, by the followng equaton. Pr r 1 = c + 1 ( c ) + Pr( = ) ( c ) 1 Pr The probabltes are, as descrbed above, calculated usng ether the bnomal or hypergeometrc dstrbutons. The c=0 Samplng Plans Many practtoners are concerned that tradtonal lot acceptance samplng plans allow nonconformng tems n the sample. For eample, the sngle sample plan n=20, c=2, r=3 allows as many as two nonconformng tems n the sample. One soluton s the use of plans that don t allow any nonconformng tems. One eample of plan s n=20, c=0, r=1. The consumer would reject the lot f any nonconformng tems appeared n the sample. 2 2 The OC Curve of Attrbute Acceptance Plans Page 6 of 7
7 Samplng plans wth c=0 don t have the same knd of OC curve dscussed above. Instead of the classc S shape, that starts to appromate the deal curve, c=0 OC curves drop off sharply wthout the bend. Fgure 6 shows the OC curves for these two plans. Notce how quckly the c=0 plan drops off. The fgure also has dashed horzontal lnes at 5% and 95% probablty of acceptance. The AQL and RQL for these plans are lsted below. c=0 c=2 AQL 0.26% 4.21% RQL 13.9% 28.3% It s easy to see that c=0 plan wll accept many fewer lots than the correspondng c=2 plan. If your process cannot tolerate even a few nonconformng unts, c=0 plans may be a good approach. However, recognze that lot rejecton ncurs a transacton cost, that may be hgh. The selecton s a c=0 plan s certanly an economc decson n=20, c=2 n=20, c=0 5.0% % 25.0% % % 5 Fgure 6 OC Curve comparson Showng c=0 Effect The OC Curve of Attrbute Acceptance Plans Page 7 of 7
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