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 the 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 predetermined δ > 0. Furthermore, we address the problem to maximize the revenue when also a maximum number of time periods is predetermined. For this problem, we describe a dynamic program if the number of possible prices is limited, and a local search algorithm if all prices are allowed. Also, we present the integer program that models this problem. Finally, we apply the obtained algorithms in a practical study.