Producer s risk

Producer s risk n. In quality control and acceptance sampling, the probability, under a given sampling plan, of making a type-I error, that is, of reflecting a lot whose true quality is at the desired acceptable level. 

The expressions producer s risk, consumer s risk, limiting quality level (LQL) and acceptable quality level (AQL) are explained in order to tmderstand how important these criteria are to warrant a high level of quality of medicines. 

Analysis or Sample-Orientation. The formulation of requirements for Craitent Uniformity (CU) however is chosen the other way around. The pharmacopoeia has formulated in every detail, how the sampling plan has to be performed and what outcome will lead to rejection or approval of the batch investigated. These requirements are formulated analysis or sample-oriented. The requirements and the influence oti consumer s risks (the chance that the consumer gets a safe product) as well as on producer s risk (the chance that a batch is incorrectly rejected) can be derived from the procedure and laid down in the Operations Characteristic (OC). The chances of acceptance are of course dependent on quality characteristics of the investigated batch such as mean content, standard deviation of the assay and percentage outliers. The OC does not formulate quality specifications, but is only a listing of chances, either to reject or to accept, as a function of a quality characteristic. 

The system of statistical end control, as mentioned in the previous section is called Acceptance Sampling. Elements are diverse parameters such as AQL, Producer s Risk, LQL, Consumer s Risk, the method of inspection or analytical procedure and the OC curve derived from them. Typically a plan contains not only the maximum and minimum limits of the content of product or batches but also the relative frequencies by which the outcome on content may be passed (or not) and what should be done accept or reject. 

Producer s Risk is the probability that a batch with a quality equal to, or better than, the AQL will be rejected. It is equivalent to the Type 1 Error in statistical hypothesis testing and is often denoted with a. 

Worked Example. A regional supplier of pharmacy preparations routinely prepares an oral solution of noscapine hydrochloride. The label claim is 100 % and the standard deviation of the spectrophotometric determination for the release decision, o = 0.8 %, is known from historic data. The limiting quality levels for this product have been fixed at LQL = 86 % with a consumer s risk, a, of about 5 %, and the AQL is set to 95 % with a producer s risk, p, of about 10 %. 

The OC curves in Fig. 20.5 are for two sample sizes, n = 3 and n = 7. The first generates too large a consumer s risk (0.14) and a producer s risk of 0.09. The second (n = 7) provides the values asked for consumer s risk 0.05 and producer s risk 0.09. Since the number of replicates is discrete, the producer s risk 0.09 is somewhat smaller than the required value of 0.10. For the same reason LQL equals only approximately the left-tailed limit of the one-sided 95 % confidence interval of p. 

Industry has accepted a default release specification for the active substance of the label claim 5 %. This would imply that in the long run, given a label claim of 100 %, a real content between 95 % and 105 % is warranted in a pre-specified proportion of the products at release. This proportion is called producer s risk and should be between 5 % and 10 % (in the example 10 %) as is usually accepted within the Statistical Quality Control (SQC) community. The corresponding content limits are thus identical to the acceptable quality level or AQL as defined above. The producer may propose to loosen these limits, when the active substance is e.g. hygroscopic, electrostatic or otherwise difficult to handle, or when the active substance is degrading considerably within the shelf life permitted. 

The values of AQL = 1 % respectively LQL = 5 % are established or agreed upon by producer and buyer of the product and can be read from the graph, while 10 % is assumed to be the target for producer s risk and consumer s risk. As explained in the previous Sect. 20.4.4 the QC has to be designed in such a way that the curve passes through points (1, 0.90) and (5, 10) as best as possible. 

An AQL of 1 % defects in the population has been chosen in this example. The producer s risk is then extremely high, namely about 25 % for a sample size n = 30 and will increase even more for larger sample sizes. Apparently it is not possible to obtain the desired acceptance plan by manipulating the sample size. The following example shows how to improve the characteristic and get a better result by changing the acceptance criterion as well. 

We would like to find an OC curve in which the consumer s risk remains small, while the producer s risk decreases. Possibly, around the AQL of 1 % defectives in the population the acceptance chance can be increased by accepting 0 or 1 defects in the sample while staying at an AQL level of about 1 %. 

This (21 %) is still above the acceptance probability of 10 % agreed upon, but more options are not available in this particular case. If an AQL of 0.5 % instead of 1 % would have been agreed upon, the producer s risk would have been 7 % instead of 21 %, which complies with the original agreement. But then the producer would in the long run have been obliged to supply a product with maximal 0.5 % defective units. Probably he did not consider this a wise option. These kinds of considerations will not be addressed... 

The concept of risk of the Bayesian theory based reliability qualification test plan is different from that of the classical qualification test plan. According to the posterior distribution, the producer s risk and user s risk are called posterior risk. (Ming 2009, Jiang Zhang 2000, Chen et al. 2002). The reliability qualification test plan based on posterior risk often makes the user bear too much risk. 

The maximum producer s risk in the test (the probability of which the product s real failure rate is smaller than the specified value but the product has failed to pass the qualification test) is,... 

By solving Equation (19), we can get a preliminary test plan T, c ) which satisfy the requirement of producer s risk a, user s risk y and the distinguishing ratio D. 

Let Ac denote the allowable cumulative degradation level of np samples corresponding to the required reliability. If CD(tp) Ac, the reliability will be accepted, otherwise the reliability will be rejected. So the customer s risk and the producer s risk associated with the demonstration above are as follows. 

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