Optimal adaptive control: A multivariate model for marketing applications

Marketing applications motivate a specialized model and its adaptive control. Each of
r
control variables is set in each of a sequence of time periods. The process being controlled has a response (profit) function that is the sum of a constant plus linear and quadratic forms in the control variables. The coefficients of the quadratic form are assumed to be known constants, those of the linear form to change with time as first-order autoregressive processes. Information about the changing coefficients is collected by performing a 2
^r
factorial experiment on a subportion of the process being controlled. Provision is made for adding further information from unspecified sources. Bayesian methods update the distributions of the unknown coefficients. Dynamic programming determines the values of the control variables and experimental design parameters to maximize the sum of discounted future profits. The probabilistic assumptions of the model are chosen so that all distributions are normal with known variances and, for the most part, zero covariances between variables. Partly as a result of this, optimal control turns out to involve rather simple exponential smoothing rules.