Our client, a manufacturer of consumer goods, became aware through customer feedback that a batch of one of their products contained an unusually high level of defects. The manufacturer wanted to take steps to avoid a repeat of this incident, but with full inspection of every product being prohibitively expensive, what could they put in place to reduce the risk of future bad batches entering the market?
The answer is a process known as acceptance sampling: inspecting samples of products against predetermined criteria in order to decide whether or not to accept batches as conforming to a certain standard. Select were asked to help set up a statistically rigorous acceptance sampling program so that the manufacturer could be confident that the defect rate in batches entering the market is acceptably low, while at the same time minimising inspection costs.
The key elements of an acceptance sampling plan are the sample size (how many products must be inspected?) and the acceptance criteria (what is the maximum number of defects that may be observed in a sample before rejection?). In order to determine these quantities we need to consider two types of error that can occur:
- A batch with an unacceptable defect rate is accepted. The probability of this happening is often called the consumer risk, since it is the consumer that would be exposed to an unacceptably high defect rate.
- A batch with an acceptable defect rate is rejected. The probability of this happening is often called the producer risk, since it is the producer that foots the bill for wasting a batch that should have been passed.
The terms “consumer risk” and “producer risk” are perhaps misnomers, since both types of error are likely to have knock-on effects for both the consumer and the producer. For example, incorrectly accepted batches may damage the producer’s reputation. On the other hand, the costs of incorrectly rejected batches may eventually get passed on to the consumer. However, the terminology is useful in terms of thinking about the problem from two competing points of view. In this case the manufacturer wanted to limit the probabilities of both types of error, and the task for Select was to find a sampling plan that achieved this trade-off.
We worked with the manufacturer to determine appropriate acceptable and unacceptable defect rates and then performed the calculations to determine the smallest sample size for which it is possible to achieve given risk levels, along with the corresponding acceptance criteria. The result was a full step-by-step sampling plan that could be implemented by quality assurance staff in the factory.
Implementation of a rigorous acceptance sampling plan has allowed the manufacturer to make precise statements about their confidence in batches achieving specified quality levels. For example, for an accepted batch, they might state:
“We are 99% confident that the defect rate in this batch is better than the acceptable level.”
On the other hand, if a batch is rejected then this action can be justified with a statement such as:
“We are 80% confident that the defect rate in this batch exceeds the unacceptable level.”
The analysis performed by Select supports statements such as these while minimising the costs associated with product inspection.