Optimal marketing decisions in a micro-level framework
A number of continuous models to explain the influence of some parameters (e.g. advertising) on the diffusion of an innovation have been proposed since the seminal paper by Bass (1969). Only some recent papers deal with both spatial and temporal features as, i.e., De Cesare et al. (2003). There the dynamic of the adopters is described by a nonlinear partial integro-differential equation. In this paper a different approach is performed. A micro-level stochastic model is built up to follow the individual paths of the potential and actual adopters. At first the influence of the information about the innovation given by local interactions is suitably treated. This leads to a Markov process describing the dynamic of adopters. Some convergence results to the continuous related models are proved. The way how marketing mix variables (advertising, prices, ...) affect the diffusion process is investigated by incorporating related parameters. Furthermore some optimal control problems are stated in order to compute the optimal marketing decision variables for a monopolistic firm's maximization profits. Here the expected value of the adopters represents the state variable whose dynamics is given through the Markov process introduced above. Due to the nonconvexity of the objective functional, random search algorithms are more appropriate because they impose few restrictions. Special attention is paid to compare the performances of simulated annealing scheme and genetic algorithms
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