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# Average Outgoing Quality Limit (AOQL) Attributes Double Sampling Plan

Double sampling plans generally require less sampling than single sampling plans but are more complicated to use.

For a double sampling plans you take a first sample consisting of n items and each item is checked to see if it is defective. The number of defective items is then counted and compared with the first sample acceptance value for the sampling plan. If the number of defectives is less or equal to the first sample acceptance value the batch is rejected. If it exceeds the SECOND sample acceptance value the batch is rejected. If the number of defectives of the first sample are more than the first sample acceptance value and less or equal to the resample value the batch is resampled. If the total number of defectives of BOTH samples exceeds the second sample acceptance value the batch is rejected and accepted otherwise.

This type of sampling plan is normally used on batches or lots prior to shipping to a customer, to ensure that the outgoing quality matches the specified level for customer acceptance. Several samples are taken randomly, and the number of defectives counted. If the number of defectives exceed a certain number for that sample size the batch is rejected, and 100 percent inspection carried out. Defective items are then removed and replaced with good items.

To understand AOQL sampling plans requires understanding some basic definitions and concepts.

Lots or batches (used interchangeably below) for Attributes Sampling Plans consist of discrete items.

An AOQL sampling plan is a rectification sampling plan.

AQL is called the Acceptable Quality Level. Some may understandably object to the term Acceptable Quality Level. There should be no such a thing as Acceptable Quality Level some will argue. The only Acceptable Quality Level some say is zero percent defectives according to critics of this concept. However, in the real world many processes cannot consistently produce products without defectives. Each process, it may be argued produces an inherent percentage of defectives, which must be accepted as a ‘fact of life’, until engineering efforts lower the average percent of defectives. Once the inherent process percent of defectives is known, it becomes the AQL.

High AQL must not be confused with high quality. It means that the average percent of acceptable level of defectives in a batch is high and hence of inferior quality, not high.

Average Outgoing Quality is the average percentage of defectives in lots which is released to the customer. Like the above, a high AOQ does not mean high quality but inferior quality.

With a rectification sampling plan the average AOQ is lower than the average percent of defectives prior to sampling. The reason is that some batches will be rejected, and hundred percent inspected and then rectified. The zero defectives in the rectified batches lowers the average percent of defectives in all released outgoing lots.

For rectification sampling plans there is a maximum AOQ over all possible percentages of defectives. Consider a low level of percentage of defectives ready to be released to the customer. If this level is less than the AQL chances are that most lots will be accepted. The outgoing quality level will be low because it is already low. Sometimes due to sampling error, even good batches will be rejected. These batches will be rectified and released with zero defectives, lowering the AOQ even further. At the other extreme consider processes with a very high level of defectives which will always be rejected by the sampling plan. Every lot will then be rejected, and hundred percent inspected and rectified. The AOQ will thus be virtually zero percent of defectives. Hence at both extremes the AOQ will be very low with a maximum somewhere in between. This maximum is called the Average Outgoing Quality Level. Whatever the initial quality level is, the outgoing quality can never exceed the AOQL due to rectification. Referring to the image below the olive colored curve with a maximum is the AOQ curve. Its peak is the AOQL. Its value can be seen on the right-hand axis and in this instance is equal to 9.7%

An AOQL sampling plan is obtained by first specifying the AOQL, which must be negotiated with the customer. The AQL must also be specified, which will be based on the current in control process. Many combinations of sample size and acceptance value will result in the specified AQL. The chosen sampling plan will be the one which results in the minimum number of items inspected for lots at the AQL and hence the need to specify the AQL.

Obtaining the sampling plan was historically achieved through optimizations. Today using machine power, approximations are no longer needed. BISNET Acceptance Sampling uses Machine Power to obtain exact probabilities of acceptance and to find the optimum sampling plan.

The output includes an OC curve as shown below, which can be used to determine the probability that a lot will be accepted or rejected at a hypothetical quality level. For example, referring to the image below, at the an assumed 18.2% of defectives there is an equal chance of accepting or rejecting the batch.

The sampling plan itself is displayed in a table.

The sample sizes and acceptance values define the plan. The AOQL is the actual AOQL achieved by this plan. Due to the discreteness of the sampling plan the exact values will not always equal the specified plan.

The producer’s risk is the risk to the producer of falsely rejecting lots at the AQL.

The Expected Items Inspected is the average number of items inspected when lots are at the AQL. This includes items inspected for lots rejected which are then 100 percent inspected.

The Single Plan EII is the expected items inspected for the equivalent single sampling plan, which should be higher.

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