6.
Process or Product Monitoring and Control
6.2. Test Product for Acceptability: Lot Acceptance Sampling


Sequential Sampling  Sequential sampling is different from single, double or multiple sampling. Here one takes a sequence of samples from a lot. How many total samples looked at is a function of the results of the sampling process.  
Itembyitem and group sequential sampling  The sequence can be one sample at a time, and then the sampling process is usually called itembyitem sequential sampling. One can also select sample sizes greater than one, in which case the process is referred to as group sequential sampling. Itembyitem is more popular so we concentrate on it. The operation of such a plan is illustrated below:  
Diagram of itembyitem sampling 


Description of sequentail sampling graph  The cumulative observed number of defectives is plotted on the graph. For each point, the xaxis is the total number of items thus far selected, and the yaxis is the total number of observed defectives. If the plotted point falls within the parallel lines the process continues by drawing another sample. As soon as a point falls on or above the upper line, the lot is rejected. And when a point falls on or below the lower line, the lot is accepted. The process can theoretically last until the lot is 100% inspected. However, as a rule of thumb, sequentialsampling plans are truncated after the number inspected reaches three times the number that would have been inspected using a corresponding single sampling plan.  
Equations for the limit lines  The equations for the two limit lines are functions of the parameters \(p_1\), \(\alpha\), \(p_2\), and \(\beta\). $$ \begin{eqnarray} x_a & = & h_1 + s n \\ x_r & = & h_2 + s n \, , \end{eqnarray} $$ where $$ \begin{eqnarray} k & = & \mbox{log} \left[ \frac{p_2(1p_1)}{p_1(1p_2)}\right] \\ h_1 & = & \frac{1}{k} \left[ \mbox{log} \left(\frac{1\alpha}{\beta}\right) \right] \\ h_2 & = & \frac{1}{k} \left[ \mbox{log} \left(\frac{1\beta}{\alpha}\right) \right] \\ s & = & \frac{1}{k} \left[ \mbox{log} \left(\frac{1p_1}{1p_2}\right) \right] \, . \end{eqnarray} $$ Instead of using the graph to determine the fate of the lot, one can resort to generating tables (with the help of a computer program).  
Example of a sequential sampling plan 
As an example, let \(p_1 = 0.01\), \(p_2 = 0.10\), \(\alpha = 0.05\),
and \(\beta = 0.10\).
The resulting equations are:
$$ \begin{eqnarray}
x_a & = & 0.939 + 0.04 n \\
x_r & = & 1.205 + 0.04 n \, .
\end{eqnarray} $$
Both acceptance numbers and rejection numbers must be integers. The
acceptance number is the next integer less than or equal to \(x_a\)
and the rejection number is the next integer
greater than or equal to \(x_r\).
Thus for \(n=1\),
the acceptance number is 1, which is impossible, and the
rejection number is 2, which is also impossible. For \(n = 24\),
the acceptance number is 0 and the rejection number is 3.
The results for \(n = 1, \, 2, \, 3, \, \ldots, \, 26\) are tabulated below.
So, for \(n=24\) the acceptance number is 0 and the rejection number is 3. The "x" means that acceptance or rejection is not possible. Other sequential plans are given below.
The corresponding single sampling plan is (52,2) and double sampling plan is (21,0), (21,1). 

Efficiency measured by ASN  Efficiency for a sequential sampling scheme is measured by the average sample number (ASN) required for a given Type I and Type II set of errors. The number of samples needed when following a sequential sampling scheme may vary from trial to trial, and the ASN represents the average of what might happen over many trials with a fixed incoming defect level. Good software for designing sequential sampling schemes will calculate the ASN curve as a function of the incoming defect level. 