Optimal multivariate bounded adjustment

A bounded adjustment strategy is an important link between statistical process control and engineering process control (or closed-loop feedback adjustment). The optimal bounded adjustment strategy for the case of a single variable has been reported in the literature and recently a number of publications have enhanced this relationship (but still for a single variable). The optimal bounded adjustment strategy for a multivariate processes (of arbitrary dimension) is derived in this article. This uses optimization and exploits a symmetry relationship to obtain a closed-form solution for the optimal strategy. Furthermore, a numerical method is developed to analyze the adjustment strategy for an arbitrary number of dimensions with only a one-dimensional integral. This provides the link between statistical and engineering process control in the important multivariate case. Both infinite- and finite-horizon solutions are presented along with a numerical illustration.