Acceptance Sampling Plan (binomial) Sample 1
Sample size(s) 51
Acc. Number(s) 5
Rej. Number(s) 6
Plan CANNOT meet desired risk point(s):
Quality RP P(accept) Plan P(accept)
PRP 0.05 0.95 0.9589318
CRP 0.15 0.20 0.2032661
To make the OC curve steeper and closer to the customer’s ideal, the required RQL can be made closer to the AQL. For example in the R code below, the RQL is reduced from 0.15 to 0.08. As a result, the find.plan function finds a plan with a much higher sample sizen= 226 (nearly 50% of the lot sizeN = 500), and acceptance numberc= 15.
R>library(AcceptanceSampling)
R>find.plan(PRP=c(0.05,0.95),CRP=c(0.08,0.20), type="hypergeom",N=500)
$n [1] 226
$c [1] 15
$r [1] 16
The OC curve for this plan is shown inFigure 2.4, and it is steeper with a reduced operating ratio. The disadvantage to this plan over the original (n= 51,c = 5) plan is the increased sample sizen= 226. The next section discusses double and multiple sampling plans that can produce a steep OC curve with a smaller average sample size than required by a single sample plan.
FIGURE 2.4: Operating Characteristic Curve for the plan with N = 500, n= 226,c= 15
A double sampling plan consists of taking a first sample of size n1. If there arec1 or less nonconforming in the sample, the lot is accepted. If there arer1 nonconforming or more in the sample, the lot is rejected (where r1 ≥ c1+ 2). If the number nonconforming in the first sample is between c1+ 1 andr1−1, a second sample of sizen2 is taken. If the sum of the number of nonconforming in the first and second samples is less than or equal toc2, the lot is accepted. Otherwise, the lot is rejected. Although there is no function in theAcceptanceSamplingpackage in R for finding double sampling plans, the assess()function and theOC2c()function can be used to evaluate a double sampling plan, and the AQLSchemes package can retrieve double sampling plans from the ANSI/ASQ Z1.4 Standard.
Consider the following example shown by Schilling and Neubauer[84]. If a single sampling plan that hasn= 134, andc= 3 is used for a lot ofN = 1000, it will have a steep OC curve with a low operating ratio. The R code below shows that there is at least a 96% chance of accepting a lot with 1% or less nonconforming, and less than an 8% chance of accepting a lot with 5% or more nonconforming.
R>library(AcceptanceSampling)
R>plns<-OC2c(n=134,c=3,type="hypergeom", N=1000, pd=seq(0,.20,.01))
R>assess(plns,PRP=c(.01,.95),CRP=c(.05,.10))
Acceptance Sampling Plan (hypergeom) Sample 1
Sample size(s) 134
Acc. Number(s) 3
Rej. Number(s) 4
Plan CAN meet desired risk point(s):
Quality RP P(accept) Plan P(accept)
PRP 0.01 0.95 0.96615674
CRP 0.05 0.10 0.07785287
However, the sample size (n=134) is large, over 13% of the lot size. If a double sampling plan with n1 = 88, c1 = 1, r1 = 4, and n2 = 88, c2 = 4, r2= 5 is used instead, virtually the same customer risk will result, and slightly less risk for the producer. This is illustrated by the R code below.
R>library(AcceptanceSampling)
R>pln3<-OC2c(n=c(88,88),c=c(1,4),r=c(4,5),type="hypergeom", N=1000,pd=seq(0,.20,.01))
R>assess(pln3,PRP=c(.01,.95),CRP=c(.05,.10)) Acceptance Sampling Plan (hypergeom)
Sample 1 Sample 2
Sample size(s) 88 88
Acc. Number(s) 1 4
Rej. Number(s) 4 5
Plan CAN meet desired risk point(s):
Quality RP P(accept) Plan P(accept)
PRP 0.01 0.95 0.9805612
CRP 0.05 0.10 0.0776524
In the code above, the first argument to theOC2c()function,n=c(88,88) specifies n1 and n2 for the double sampling plan. The second argument c=c(1,4)specifies c1 and c2, and the third argument r=c(4,5) specifiesr1 and r2. Notice that r2 = c2+ 1 because a decision must be made after the second sample.
The sample size for a double sampling plan will vary betweenn1andn1+n2 depending on whether the lot is accepted or rejected after the first sample.
The probability of accepting or rejecting after the first sample depends upon the number of nonconforming items in the lot, therefore the average sample number (ASN) for the double sampling plan will be:
ASN =n1+n2×P(c1< x1< r1) (2.2)
wherex1is the number of nonconforming items found in the first sample.
The R code below createsFigure 2.5that compares the sample size for the single sampling plan with the average sample for a double sampling plan at various proportions nonconforming or defective in the lot.
R>library(AcceptanceSampling)
R>D=seq(0,200,5) #Number nonconforming in the lot of 1000 R>pd<-D/1000 #Proportion nonconforming in the lot of 1000 R>OC1<-phyper(1, m=D, n=1000-D, k=88, lower.tail=TRUE) R>#Probability of accepting after the first sample R> R1<-phyper(3, m=D, n=1000-D, k=88, lower.tail=FALSE) R>#Probability of rejecting after the first sample R>P<-OC1+R1
R>ASN=88+88*(1-P)
R>plot(pd,ASN,type=’l’,ylim=c(5,150),xlab="Proportion nonconforming in the lot")
R>abline(134,0,lty=2)
R>text(.10,142,’single sample n=134’) R>text(.10,70,’double sample n=(88,88)’) R>grid()
FIGURE 2.5: Comparison of Sample Sizes for Single and Double Sampling Plan
From this figure it can be seen that the average combined sample size for the double sampling plan is uniformly less than the equivalent single sampling
plan. The average sample size for the double sampling plan saves most when the proportion nonconforming in the lot is less than the AQL or greater than the RQL.
The disadvantage of a double sampling plan is that they are very difficult or impossible to apply when the testing or inspection takes a long time or must be performed off site. For example, food safety and microbiological tests may take 2 to 3 days for obtaining the result.
Multiple sampling plans extend the logic of double sampling plans, by further reducing the average sample size. Multiple sampling plans can be presented in tabular form as shown inTable 2.1
If x1 ≤ c1 (where x1 is the number of nonconforming items found in the first sample) the lot is accepted. If x1 ≥ r1 the lot is rejected, and if c1< x1< r1 another sample is taken, etc.
A multiple sampling plan with a similar OC curve as a double sampling plan will have an even lower ASN curve than the double sampling plan. The multiple sampling plan shown inTable 2.2, has an OC curve that is very similar to the single (n=134, c=3) and the double sampling plan (n1=88, n2=88, c1=1, c2=4,r1=4,r2=5) presented above.Figure 2.6shows a comparison of their OC curves magnifying on the region where these curves are steepest. It can be seen that within the region of the AQL=0.01 and the RQL = 0.05, these OC curves are very similar. The ASN curve for the multiple sampling plan can be shown to fall below the ASN curve for the double sampling plan shown inFigure 2.5.
Although the AcceptanceSamplingpackage does not have a function for creating double or multiple sampling plans for attributes, the ANSI/ASQ Z1.4 tables discussed in Section 2.7.2 present single, double, and multiple sampling plans with matched OC curves. The single, double, and multiple sampling plans in these tables can be accessed with theAQLSchemespackage or with the [sqc online calculator](https://www.sqconline.com/). The tables also present the OC curves and ASN curves for these plans, but the same the OC and ASN curves can be obtained from theAQLSchemespackage as well.
TABLE 2.1: A Multiple Sampling Plan
Sample Samp. Size Cum. Sample Size Accept. Number Reject. Number
1 n1 n1 c1 r1
2 n2 n1+n2 c2 r2
... ... ... ... ...
k nk n1+n2+. . .+nk ck rk=ck+ 1
TABLE 2.2: A Multiple Sampling Plan with k=6
Sample Samp. Size Cum. Samp. Size Accept. Number Reject. Number
1 46 46 0 3
2 46 92 1 3
3 46 138 2 4
4 46 184 3 5
5 46 230 4 6
6 46 276 6 7
FIGURE 2.6: Comparison of OC Curves
TheOC2c(), andassess()functions in theAcceptanceSamplingpackage, and the plot() function in R can be used to evaluate the properties of a multiple sampling plan as shown in the R code below.
R>library(AcceptanceSampling)
R>pln4<-OC2c(n=c(46,46,46,46,46,46),c=c(0,1,2,3,4,6), r=c(3,3,4,5,6,7),
type="hypergeom",N=1000,pd=seq(0,.20,.01)) R>assess(pln4,PRP=c(.01,.95),CRP=c(.05,.10))
R>plot(pln4,type=’l’) R>summary(pln4,full=TRUE)
In summary, a sampling plan with a steeper OC curve is preferable. It benefits the customer by reducing the chances of accepting lots containing a proportion nonconforming higher than the AQL, and it motivates the supplier to send lots with a proportion nonconforming less than the AQL to avoid having lots returned. It also benefits a supplier of good quality because the probability of having a lot rejected that contains a proportion nonconforming less than the AQL is reduced.
Single sampling plans are the easiest to administer; however, the disadvan- tage of a single sampling plan with a steep OC curve is the additional cost of increased sampling. An equivalently steep OC curve can result from a double or multiple sampling plan with a lower average sample number (ASN). This is the advantage of double and multiple sampling plans. The disadvantage of double or multiple sampling plans is the increased bookkeeping and additional decisions that must be made.