Economic lot-sizing with remanufacturing: complexity and efficient formulations

Within the framework of reverse logistics, the classic economic lot-sizing problem has been extended with a remanufacturing option. In this extended problem, known quantities of used products are returned from customers in each period. These returned products can be remanufactured so that they are as good as new. Customer demand can then be fulfilled from both newly produced and remanufactured items. In each period, one can choose to set up a process to remanufacture returned products or produce new items. These processes can have separate or joint setup costs. In this article, it is shown that both variants are NP-hard. Furthermore, several alternative mixed-integer programming (MIP) formulations of both problems are proposed and compared. Because “natural” lot-sizing formulations provide weak lower bounds, tighter formulations are proposed, namely, shortest path formulations, a partial shortest path formulation, and an adaptation of the (l, S, WW) inequalities used in the classic problem with Wagner–Whitin costs. Their efficiency is tested on a large number of test data sets and it is found that, for both problem variants, a (partial) shortest path–type formulation performs better than the natural formulation, in terms of both the linear programming relaxation and MIP computation times. Moreover, this improvement can be substantial.