One of the most widely used quality control tools is the attribute acceptance sampling plan, which can be applied in a variety of ways. For example, in the context of manufacturing, it can be used to make sure that the quality of incoming parts satisfies certain requirements before they are assembled, that the quality of semi-finished products is acceptable before they are passed to the next manufacturing stage, or that the quality of finished products satisfies the customer’s specifications before they are shipped.
Each attribute sampling plan has three parameters (N, n, c) -- lot size, sample size, and acceptance number, respectively. The operation of an attribute sampling plan is simple. You select a random sample of n units from the incoming lot of size N. You then determine the number of defective components in the sample. If this number does not exceed the pre-determined c, the lot is accepted; otherwise the lot is rejected.
The quality level of a lot is usually expressed as percentage defective or fraction defective. For instance, a quality level of p = 10% means that there are 10% defective units in the lot. If p = 0%, all units in the lot are good, whereas if p = 100%, all units are defective. It is worth noting that, if a lot has a quality level of p = 0%, then it will be accepted no matter what sampling plan (N, n, c) is chosen. However, unless the quality level p is 100%, a lot with a high percent defective, say 30%, will still have a chance of being accepted.
Despite its wide applicability, there are two types of risk associated with each attribute sampling plan.
The risks associated with a sampling plan can be read from the operating characteristic (OC) curve, which is a graph showing the probability Pa of lot acceptance for different lot quality levels.
There are two ways to calculate the probability of lot acceptance. Suppose that the sampling plan is (N, n, c) and the quality level of the lot is p. The first method is an exact one. The number X of defectives found in a sample will follow a Hypergeometric distribution, so the probability of lot acceptance is
Note that Np is an integer in this formula.
The second method is approximate. When the ratio is small, the Hypergeometric distribution can be approximated well by the Binomial distribution. Thus the probability of acceptance can be approximated by
The following figure shows the OC curves for plans with N = 1000, n = 50, c = 0 and N = 500, n = 25, c = 0. Here the probability of acceptance is calculated using the Hypergeometric distribution.
Using the plan (N, n, c) = (500, 25, 0), a lot with a relatively good quality level of 0.01 will still have about a chance of 0.23 being rejected. That is, the producer’s risk is 0.23. Using the same plan, a lot with a relatively bad quality level of 0.1 will still have a chance of 0.07 being accepted. That is, the consumer’s risk is 0.07. It is worth noting that, despite the two plans (500, 25, 0) and (1000, 50, 0) having the same ratio, their OC curves are different.