Bayesian Sampling Plans for Inspection by Variables

We select an appropriate sampling plan from a sampling inspection system for inspection by variables, i.e. ISO 3951, and base the lot acceptance decision and the adaptation of the sampling plan directly on the a posteriori distribution of the fraction of nonconforming items in the lots and especially the a posteriori estimate of the probability of the fraction of nonconforming items in the lot being larger than the acceptance quality limit AQL. We do not assume a prior distribution of the fraction nonconforming in the lots because the production process does not directly generate fractions of nonconforming items but items with a quantitative characteristic assumed to be normally distributed with parameters μ (lot mean) and σ² (within-lot variance) varying in time, respectively from lot to lot. The process curve, i.e. the two-dimensional distribution of the lot means and the within-lots variances is assumed to be Normal-scaled-inverse-chi-squared. In our hierarchical Bayes model we estimate the parameters of the process curve directly by exponentially weighted means and variances of the sample averages and the sample variances of the already inspected lots. Switching between tightened, normal and reduced inspection turns out to be more straightforward than with the switching rules of ISO 3951. Furthermore, if the process curve of the variance is stable, it is possible to switch from sampling plans with unknown variance to sampling plans with known variance. We apply the Bartlett test for switching to σ-plans and a CUSUM-s ²-chart for switching back to s-plans.