  A bivariate extension to Box and Jenkins (1963) feedback adjustment problem
is presented in this paper. The model balances the fixed cost of making an
adjustment, which is assumed independent of the magnitude of the adjustments,
with the cost of running the process off-target, which is assumed quadratic. It
is also assumed that two controllable factors are available to compensate for
the deviations from target of two responses in the presence of a bivariate
IMA(1,1) disturbance. The optimal policy has the form of a "dead band", in
which adjustments are justified only when the predicted process responses
exceed some boundary in $\mathbb{R}^2$. This boundary indicates when the
responses are predicted to be far enough from their targets that an additional
adjustment or intervention in the process is justified. Although originally
developed to control a machine tool, dead band control policies have
application in other areas. For example, they could be used to control a
disease through the application of a drug to a patient depending on the level
of a substance in the body (e.g., diabetes control). This paper presents
analytical formulae for the computation of the loss function that combines
off-target and adjustment costs per time unit. Expressions are derived for the
average adjustment interval and for the scaled mean square deviations from
target. The minimization of the loss function and the practical use of the
resulting dead band adjustment strategy is illustrated with an application to a
semiconductor manufacturing process.
