Optimal control promotional policy for consumer durable product in a segmented market incorporating goodwill

Owing to the competitive structure of the market, firms concentrate excessively on promotional activities to maximize their profit. In this paper, we formulate an optimal control problem to find promotional effort policies for the case when the firm promotes its product at mass and segment specific level in a segmented market. The adoption is measured through the innovation diffusion model with respect to mass and differentiated promotion. The objective of proposed model is profit maximization where the sales evolve through promotion which in turns built up goodwill described by Nerlove-Arrow Model. The solution of the proposed problem is obtained using maximum principle. Some special cases of the proposed optimal control problem are also discussed. The application of the discretized version of the proposed problem is illustrated through a numerical.