  Quality control in industrial processes is increasingly making use of prior
scientific knowledge, often encoded in physical models that require numerical
approximation. Statistical prediction, and subsequent optimization, is key to
ensuring the process output meets a specification target. However, the
numerical expense of approximating the models poses computational challenges to
the identification of combinations of the process factors where there is
confidence in the quality of the response. Recent work in Bayesian computation
and statistical approximation (emulation) of expensive computational models is
exploited to develop a novel strategy for optimizing the posterior probability
of a process meeting specification. The ensuing methodology is motivated by,
and demonstrated on, a chemical synthesis process to manufacture a
pharmaceutical product, within which an initial set of substances evolve
according to chemical reactions, under certain process conditions, into a
series of new substances. One of these substances is a target pharmaceutical
product and two are unwanted by-products. The aim is to determine the
combinations of process conditions and amounts of initial substances that
maximize the probability of obtaining sufficient target pharmaceutical product
whilst ensuring unwanted by-products do not exceed a given level. The
relationship between the factors and amounts of substances of interest is
theoretically described by the solution to a system of ordinary differential
equations incorporating temperature dependence. Using data from a small
experiment, it is shown how the methodology can approximate the multivariate
posterior predictive distribution of the pharmaceutical target and by-products,
and therefore identify suitable operating values. Materials to replicate the
analysis can be found at www.github.com/amo105/chemicalkinetics.
