Developing a model-based sampling plan

Statistical process modeling is widely used in industry for forecasting the production outcomes, for process control and for process optimization. Applying a prediction model in a production process allows the user to calibrate/predict the mean of the distribution of the process outcomes and to partition the overall variation in the distribution of the process outcomes into explained (by the model) and unexplained (residuals) variations; thus, reducing the unexplained variability. The additional information about the process behavior can be used prior to the sampling procedure and may help to reduce the required sample size to classify a lot. This research focuses on the development of a model-based sampling plan based ontextitCpk (process capability index). It is an extension of a multistage acceptance sampling plan also based on Cpk (2009). The advantage of this sampling plan is that the sample size needed depends directly and quantitatively on the quality of the process (Cpk), whereas other sampling plans such as MIL-STD-414 (1957.) use only qualitative measures. The objective of this paper is to further refine the needed sample size by using a predictive model for the lot's expectation. We developed model-based sample size formulae which depend directly on the quality of the prediction model (as measured by R²) and adjust the ‘not model-based’ multistage sampling plan developed in Negrin et al. (2009, 2011) accordingly. A simulation study was conducted to compare between the model-based and the ‘not model-based’ sampling plans. It is found that when R² = 0, the model-based and ‘not model-based’ sampling plans require the same sample sizes in order to classify the lots. However, as R² becomes larger, the sample size required by the model-based sampling plan becomes smaller than the one required by the ‘not model-based’ sampling plan. In addition, it is found that the reduction of the sample size achieved by the model-based sampling plan becomes more significant as Cpk tends to 1 and can be achieved without increasing the proportion of the classification errors. Finally, the suggested sampling plan was applied with areal data set from a chemicals manufacturing process for illustration.