Appendix 2. B The convolution theory
3.4 Two-class compressed limit attribute plans for known σ
The termcompressed limitis synonymous with terms such as pseudo-specification, tightened limit, and narrow limit. A compressed limit is an artificial limit which is fixed well below the regulatory or specification limit. Sampling plans based on compressed limits have been studied byOtt and Mundel(1954),Beja and Ladany(1974),Schilling and Sommers(1981) andEvans and Thyregod(1985) and others.
The traditional compressed limit sampling plans are based on the normal distribution and the standard deviation is assumed to be known and stable. Log transformed microbial counts are generally assumed to be normally distributed with a known standard deviation on the log10 scale
σ =0.8 (Dahms,2004;Legan et al.,2001). This assumed batch standard deviation is larger than usually expected and therefore the consumers risk is not adversely affected. Compressed limit plans use the same decision criterion as the two-class attribute plans namelydc; where
d is the observed number of nonconforming analytical results beyond the tightened limit and
cis the acceptance number. Compressed limit plans partly take advantage of the underlying continuous probability distribution of the variable of interest and hence require smaller sample sizes. Compressed limit plans may achieve a reduction in the sample size of about 80% when compared to using uncompressed specification limit (Schilling and Neubauer,2010).
The procedure of setting a compressed limit for the normal distribution is described below. LetZpbe the quantile in the normal distribution associated with a proportion nonconforming p. See Figure3.2. By compressing the specification bytstandard deviations (σ), an artificial proportion nonconformingg results. The normal distribution quantile corresponding to this artificial proportion nonconformingZgyields the compressed limit. Therefore,
Zg=Zp−σt (3.2)
In the literaturetis known as the compression constant , and the valuet=1 is often employed for the sake of simplicity. However,t=1 compressed limit plans may not be optimal for controlling the producer’s and consumer’s risks at desired levels.
3.4 Two-class compressed limit attribute plans for knownσ 35 0.0 0.1 0.2 0.3 0.4 nor m al density m CL μ Zg σ Zp σ t σ g p
Fig. 3.2 Illustration of the compressed limit approach in the normal distribution.
For determination of the optimumt value,Ladany(1976) proposed an iterative graphical method based on the nomograph of the cumulative binomial distribution given byLarson(1966). An approximate heuristic approach was later proposed bySchilling and Sommers(1981) who provided the following formulae for the compressed limit plan parameters:
nt=1.5nv
t=k
ct=0.75nv−0.67
(3.3)
wherenv is the sample size of the variables plan andkthe critical distance obtained for the tradi-
tional variables plan, seeDuncan(1986);Schilling and Neubauer(2010). Another approximate optimal solution was suggested byEvans and Thyregod(1985). This method is based on the normal approximation to the binomial distribution and yields better results. The formulae for the design of compressed limit plans are
nt =π 2 Z α+Zβ ZAQL−ZLQL 2 t =k= ZαZLQL+ZβZAQL Zα+Zβ ct = (nt−1)/2 (3.4)
Sampling plans published inSchilling and Sommers(1981) are not exact since the risks were relaxed by allowing tolerancesα+0.005 andβ+0.005 which leads to smaller sample sizes. Other approaches obtained by approximation to the binomial model lead to slightly different results. Hence we provide below a new algorithm to obtain the exact optimal compressed limit plans.
1. Given two points (AQL,α) and (LQL,β), compute the corresponding standard normal quantilesZAQLandZLQL.
2. For the sequence of t =0(0.01)4, calculate the normal quantiles Zg1 =ZAQL−t and
Zg2=ZLQL−t.
3. Obtain the artificial proportions nonconformingpg1 and pg2 as the right tail areas of the
standard normal distribution corresponding toZg1 andZg2.
4. For given pair of points (pg1,α) and (pg2,β)of the OC curve, obtainnt andct by solving
the binomial inequalities:
1−α= c
∑
d=0 n d pdg11−pg1 n−d 1−α β=∑
c d=0 n d pdg21−pg2 n−dβ (3.5)wheredn=n!/(d!(n−d)!) is the binomial coefficient, ! is the factorial andn,c and
dare nonnegative integers.α andβ are the predefined producer’s and consumer’s risks andα andβ are the achieved producer’s and consumer’s risks. Note that Eq.3.5implies
αα andβ β.Guenther(1969), for instance, provides an algorithm to solve these inequalities.
5. Select thet value that minimizes the sample size.
6. When more than one sampling plan exists, a second optimality criterion has to be em- ployed. We propose the criterion 1 of the maximum absolute risk difference (MARD) maxα−α+β−β
. The MARD criterion provides a slightly tighter OC curve than desired whenα >α andβ >β, and hence the designed plan is more stringent .
Other alternatives to the MARD criterion are available in the literature, for instance, the minimum absolute risk difference (MIRD) minα−α+β−β
Schilling and Sommers
(1981). The method proposed inSchilling and Sommers(1981) does not impose the conditions
α >α andβ >β and hence it gives the closest OC curve to the points (AQL,α) and (LQL,β).
Evans and Thyregod(1985) suggested the use of the midpoint between all the possibletvalues. Appendix3.Cgives the compressed limit plans based on two points on the OC curve. The sampling design is also given for the MIRD criterion. TheLQLvalues correspond to the selected
3.4 Two-class compressed limit attribute plans for knownσ 37
operating ratiosR=LQL/AQLequal to 20 and 40. For very smallAQLvalues,t tends to be large which may cause a conflict with the specification limit.3.Bcontains the R codes to obtain the optimal sampling design for other combinations of risks. We have also built an easy-to-use web application (app) using theR packageshiny(Chang et al.,2015) for those practitioners unfamiliar withR. This tool is available at https://edgarsantosfdez.shinyapps.io/compress. As seen in Appendix3.C, when bothAQLandLQLare very low, the usual uncompressed attribute plans require large sample sizes such asn=313. We show that a substantial reduction in the sample size can be achieved using the optimum compressed limit.
The following step-by-step guide illustrates the design and operation of the compressed limit attribute plan:
1. Assess the fit to a normal distribution and the stability of the variance using control charts.
2. Using two points (AQL,α) and (LQL,β), obtain the number of samples to be drawn (nt),
the acceptance number (ct) and the quantile (qt) from3.C. For other quality levels and/or
risks, use the R codes given in3.B.
3. Obtain the artificial limitCLas theqt quantile of the normal distribution.
4. Inspect thent items and determine the number of artificially nonconforming items (dt) for
theCLlimit.
5. Ifdt ct, accept the batch; otherwise reject.
Zero acceptance number sampling plans
Zero acceptance number (c=0) inspection plans are desired in several industrial applications. In food safety, the plans are generally designed using one point in the OC curve (LQL, β) plus the restriction c=0. Here, the producer’s point (AQL, α) is not relevant because for food safety assurance the batch is expected to be free of pathogens. In the inspection of pathogenic microorganisms, it is not possible to release a lot when one of the samples fails the microbiological limit. Therefore the compressed limit plans introduced earlier should be limited toct=0. The algorithm to find the compressed limit zero acceptance number plan for the known σ case is described below.
1. Given the point (LQL,β) andct=0 compute the standard normal quantileZLQL.
2. Select a reasonable value fort, sayt=1 and obtain the normal quantileZg2 =ZLQL−t.
3. Obtain the artificial proportion nonconforming pg2 corresponding toZg2 (right tail area of
the standard normal distribution).
4. Use pg2 andβ obtainnt
nt=
log(β) log1−pg2
Two-class compressed limit for unknown
σ
For the traditional variables inspection plans, the sample size for the unknownσ case is approx- imately1+k2/2times the sample size of the knownσ plan, seeWallis(1947) orSchilling and Neubauer (2010). The compressed limit attribute plan is also expected to be sensitive to the uncertainty of the population variance. However, the design of compressed limit attribute plans when the condition of knownσ is not satisfied needs to be considered. We developed the following Monte Carlo simulation procedure to obtain the optimal compressed attribute plans for unknownσ.
1. For the normal distribution, there exists a one-to-one relationship betweenAQLand m. That is, for given producer’s point (AQL,α), obtain the specification limit as the (1−AQL)- quantile of the standard normal distribution.
2. Generate a random sample of sizenfrom the standard normal distribution (μ=0,σ =1). 3. Obtain the compressed limitCLasm−t.
4. Obtain the number of artificial nonconforming items (d) as the number of observations of the sample greater thanCL.
5. Obtain empirically the probability of acceptance as the proportion of cases in whichdc
using at least 5000 iterations.
6. Consider selectμ >0 values. Obtain the probability of acceptancePafor a range of μ
values.
7. To determine whether a given combination ofn,candt produces an OC curve restricted to the given two points, the following conditions need to be satisfied: Pa1−α andPaβ
atAQLandLQLrespectively.
8. The optimum plan is the one that minimizes the sample size and satisfies the two point restrictions.
9. When more than one sampling design exists obtain the plan applying MARD or MIRD criterion.
The sampling plans designed for common combinations of quality levels are also shown in Appendix3.C(matched with other sampling plans). The operation of this sampling plan is similar to the two-class compressed limit plan for knownσ discussed earlier.