Double-Sampling Plans for Attributes

A double sampling plan for attributes is a two-stage procedure. In the first stage, a random sample of size $n_1$ is drawn, without replacement, from the lot. Let $X_1$ denote the number of defective items in this first stage sample. Then the rules for the second stage are the following: if $X_1\leq c_1$, sampling terminates and the lot is accepted; if $X_1 \geq c_2$, sampling terminates and the lot is rejected; if $X_1$ is between $c_1$ and $c_2$, a second stage random sample, of size $n_2$, is drawn, without replacement, from the remaining items in the lot. Let $X_2$ be the number of defective items in this second-stage sample. Then, if $X_1 + X_2 \leq c_3$, the lot is accepted and if $X_1 + X_2 > c_3$ the lot is rejected.

Generally, if there are very few (or very many) defective items in the lot, the decision to accept or reject the lot can be reached after the first stage of sampling. Since the first stage samples are smaller than those needed in a single stage sampling a considerable saving in inspection cost may be attained.

In this type of sampling plan, there are five parameters to select, namely, $n_1$, $n_2$, $c_1$, $c_2$ and $c_3$. Variations in the values of these parameters affect the operating characteristics of the procedure, as well as the expected number of observations required (i.e. the total sample size). Theoretically, we could determine the optimal values of these five parameters by imposing five independent requirements on the OC function and the function of expected total sample size, called the Average Sample Number or ASN-function, at various values of $p$. However, to simplify this procedure, it is common practice to set $n_2 = 2n_1$ and $c_2 = c_3 = 3c_1$. This reduces the problem to that of selecting just $n_1$ and $c_1$. Every such selection will specify a particular double-sampling plan. For example, if the lot consists of $N=150$ items, and we choose a plan with $n_1 = 20$, $n_2 = 40$, $c_1 = 2$, $c_2 = c_3 = 6$, we will achieve certain properties. On the other hand, if we set $n_1 = 20$, $n_2 = 40$, $c_1 = 1$, $c_2 = c_3 = 3$, the plan will have different properties.

The formula of the OC function associated with a double-sampling plan $(n_1,n_2,c_1,c_2,c_3)$ is

$\text{OC}(p) = H(c_1;N,M_p,n_1) + \sum\limits_{j=c_1+1}^{c_2-1} h(j;N,M_p,n_1)H(c_3-j;N-n_1,M_p-j,n_2)$

where $M_p = [Np]$. Obviously, we must have $c_2\geq c_1 + 2$, for otherwise the plan is a single-stage plan.