Multiple item economic lot sizing problem with inventory dependent demand

We consider a multiple item Economic Lot Sizing problem where the demands for items depend on their stock quantities. The objective is to find a production plan such that the resulting stock levels (and hence demands) maximize total profit over a finite planning horizon. The single item version of this problem has been studied in the literature, and a polynomial time algorithm has been proposed when there are no bounds on production. It has also been proven that the single item version is NP-hard even when there are constant (i.e., time-invariant) finite capacities on production. We extend this single item model by considering multiple items and production capacities. We propose a Lagrangian relaxation method to find an initial solution to the problem. This solution is a hybrid solution obtained by combining two distinct solutions generated in the process of solving the Lagrangian dual problem. Starting with this initial solution, we then implement a Tabu Search algorithm to find better solutions. The performance of the proposed solution method is compared with the performance of a standard commercial software that works on a mixed integer programming formulation of the problem. We show that our solution approach finds better solutions within a predetermined time limit in general.