A sequential Markov Chain Monte Carlo approach to set-up adjustment of a process over a set of lots

We consider the problem of adjusting a machine that manufactures parts in batches or lots and experiences random offsets or shifts whenever a set-up operation takes place between lots. The existing procedures for adjusting set-up errors in a production process over a set of lots are based on the assumption of known process parameters. In practice, these parameters are usually unknown, especially in short-run production. Due to this lack of knowledge, adjustment procedures such as Grubbs' (1954, 1983) rules and discrete integral controllers (also called EWMA controllers) aimed at adjusting for the initial offset in each single lot, are typically used. This paper presents an approach for adjusting the initial machine offset over a set of lots when the process parameters are unknown and are iteratively estimated using Markov Chain Monte Carlo (MCMC). As each observation becomes available, a Gibbs Sampler is run to estimate the parameters of a hierarchical normal means model given the observations up to that point in time. The current lot mean estimate is then used for adjustment. If used over a series of lots, the proposed method allows one eventually to start adjusting the offset before producing the first part in each lot. The method is illustrated with application to two examples reported in the literature. It is shown how the proposed MCMC adjusting procedure can outperform existing rules based on a quadratic off-target criterion.