A fractional optimal control problem for maximizing advertising efficiency

We propose an optimal control problem to model the dynamics of the communication activity of a firm with the aim of maximizing its efficiency. We assume that the advertising effort undertaken by the firm contributes to increase the firm's goodwill and that the goodwill affects the firm's sales. The aim is to find the advertising policies in order to maximize the firm's efficiency index which is computed as the ratio between "outputs" and "inputs" properly weighted; the outputs are represented by the final level of goodwill and by the sales achieved by the firm during the period considered, whereas the inputs are represented by the costs undertaken by the firm, fixed costs and advertising costs. The problem considered is formulated as a fractional optimal control problem. In order to find the optimal advertising policies we use the Dinkelbach's algorithm for fractional programming.