Optimal Institutional Advertising: Minimum-Time Problem

This paper considers the problem of optimizing the institutional advertising expenditure for a firm which produces two products. The problem is formulated as a minimum-time control problem for the dynamics of an extended Vidale-Wolfe advertising model, the optimal control being the rate of institutional advertising that minimizes the time to attain the specified target market shares for the two products. The attainable set and the optimal control are obtained by applying the recent theory developed by Hermes and Haynes extending the Green's theorem approach to higher dimensions. It is shown that the optimal control is a strict bang-bang control. An interesting side result is that the singular arc obtained by the Green's theorem application turns out to be a maximum-time solution over the set of all feasible controls. The result clarifies the connection between the Green's theorem approach and the maximum principle approach.