Evaluating Single-Sampling Plans

Functions

Evaluating Single-Sampling Plans

You can use the base SAS functions PROBBNML and PROBHYPR to evaluate single-sampling plans. Measures of the performance of single-sampling plans include

the probability of acceptance P_a
the average sample number ASN
the average outgoing quality AOQ
the average total inspection ATI

Probability of Acceptance

Since P_a is the probability of finding c or fewer defectives in the sample, you can calculate the acceptance probability using the function PROBHYPR(N,D,n,c) for Type A sampling and the function PROBBNML(p,n,c) for Type B sampling.

For example, the following statements calculate P_a for the plan n=20, c=1 when sampling from a single lot of size N=120 that contains D=22 nonconforming items, resulting in a value of 0.0762970752:

   data;
      prob=probhypr(120,22,20,1);
      put prob;
   run;

Similarly, the following statements calculate P_a for the plan n=20, c=1 when sampling from a series of lots for which the proportion of nonconforming items is p=0.18, resulting in a value of 0.1018322793:

   data;
      prob=probbnml(0.18,20,1);
      put prob;
   run;

Other Measures of Performance

The measures ASN, AOQ, and ATI are meaningful only for Type B sampling and can be calculated using the PROBBNML function. For reference, the following equations are provided.

Average sample number: Following the notation of Schilling (1982), let F(c|n) denote the probability of finding c or fewer nonconforming items in a sample of size n. Note that F(c|n) is equivalent to PROBBNML(p,n,c). Then, depending on the mode of inspection, the average sample number can be expressed as shown in the following table:

Mode of Inspection ASN

Full n

Semicurtailed $nF(c| n) + \displaystyle \frac{ (c+1)(1-F(c+1| n+1)) }p$

Fully curtailed $\displaystyle \frac{(n-c)F(c| n+1)}{1-p} + \displaystyle \frac{(c+1)(1-F(c+1| n+1))}p$

Average outgoing quality can be expressed as

${AOQ}=\frac{p(N-n)F(c| n)}N$

if the nonconforming items found are replaced with conforming items, and as

${AOQ}=\frac{p(N-n)F(c| n)}{N-np}$

if the nonconforming items found are not replaced.

Average total inspection can be expressed as

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Mode of Inspection	ASN
Full	n
Semicurtailed	$nF(c\| n) + \displaystyle \frac{ (c+1)(1-F(c+1\| n+1)) }p$
Fully curtailed	$\displaystyle \frac{(n-c)F(c\| n+1)}{1-p} + \displaystyle \frac{(c+1)(1-F(c+1\| n+1))}p$