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

What is a Sequential Sampling Plan?

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.
Item-by-item and group sequential sampling The sequence can be one sample at a time, and then the sampling process is usually called item-by-item sequential sampling. One can also select sample sizes greater than one, in which case the process is referred to as group sequential sampling. Item-by-item is more popular so we concentrate on it. The operation of such a plan is illustrated below:
Diagram of item-by-item sampling

Description of sequentail sampling graph The cumulative observed number of defectives is plotted on the graph. For each point, the x-axis is the total number of items thus far selected, and the y-axis 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, sequential-sampling 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 $$\begin{eqnarray} k & = & \mbox{log} \left[ \frac{p_2(1-p_1)}{p_1(1-p_2)}\right] \\ h_1 & = & \frac{1}{k} \left[ \mbox{log} \left(\frac{1-\alpha}{\beta}\right) \right] \\ h_2 & = & \frac{1}{k} \left[ \mbox{log} \left(\frac{1-\beta}{\alpha}\right) \right] \\ s & = & \frac{1}{k} \left[ \mbox{log} \left(\frac{1-p_1}{1-p_2}\right) \right] \, . \end{eqnarray}$$ 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.

 $$n$$ inspect $$n$$ accept $$n$$ reject $$n$$ inspect $$n$$ accept $$n$$ reject 1 x x 14 x 2 2 x 2 15 x 2 3 x 2 16 x 3 4 x 2 17 x 3 5 x 2 18 x 3 6 x 2 19 x 3 7 x 2 20 x 3 8 x 2 21 x 3 9 x 2 22 x 3 10 x 2 23 x 3 11 x 2 24 0 3 12 x 2 25 0 3 13 x 2 26 0 3

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.

 $$n$$ inspect $$n$$ accept $$n$$ reject 49 1 3 58 1 4 74 2 4 83 2 5 100 3 5 109 3 6

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.