Price strategy implementation

Consider a situation in which a company sells several different items to a set of customers. However, the company is not satisfied with the current pricing strategy and wishes to implement new prices for its items. Implementing these new prices in one single step might not be desirable, for example, because of the change in contract prices for the customers. Therefore, the company changes the prices gradually, such that the prices charged to a subset of the customers, the target market, do not differ too much from one period to the next. We propose a polynomial time algorithm to implement the new prices in the minimum number of time periods needed, given that the prices charged to the customers in the target market increase by at most a factor 1+δ, for a given δ>0. Furthermore, we address the problem of maximizing the overall revenue during the price implementation over a given time horizon. For this problem, we describe a dynamic program for the case of integer price vectors, and a local search algorithm for arbitrary prices. Also, we present a mixed integer programming formulation for this problem and apply our algorithms in a practical study.