Efficient reformulations for dynamic lot-sizing problems with product substitution

We consider single-level uncapacitated and capacitated lot-sizing problems with product substitution, where products may be substituted by certain other products to satisfy demand. The models incorporate initial inventories and general substitution structures. We formulate the problems as mixed-integer linear programs and develop Simple Plant Location-based reformulations as well as new valid inequalities. Computational results on generated problem instances show that the reformulations are superior to the original formulations and those with valid inequalities added a priori, except for instances with multiple resources and downward substitution. In most cases, the running times of a mixed-integer programming solver on approximate extended formulations that only contain a subset of the disaggregated constraints were almost as good as on complete Simple Plant Location-based reformulations.