New product introduction: goodwill, time and advertising cost

The advertising activity has an important role in the introduction of a new product. We tackle the problem of determining the advertising plan for preparing the product introduction, having the objectives of maximizing the product image at the launch time, minimizing the campaign length and minimizing the total advertising expenditure, within a fixed time interval. After formulating the new product introduction optimal control problem, we study and solve two special cases of it, namely the minimum time and the advertising cost problems. From the former we derive the existence of an optimal solution of the original problem, whereas we obtain its structure from the latter. Finally we present two alternative formulations of the original problem in the non-linear programming framework and suggest using them to solve the bounded time problem.