A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

In this paper, we study a multi-echelon uncapacitated lot-sizing problem in series (mULS), where the output of the intermediate echelons has its own external demand, and is also an input to the next echelon. We propose a polynomial-time dynamic programming algorithm, which gives a tight, compact extended formulation for the two echelon case (2-ULS). Next, we present a family of valid inequalities for m-ULS, show its strength and give a polynomial-time separation algorithm. We establish a hierarchy between the alternative formulations for 2-ULS. In particular, we show that our valid inequalities can be obtained from the projection of the multi-commodity formulation. Our computational results show that this extended formulation is very effective in solving our uncapacitated multi-item 2-echelon test problems. In addition, for capacitated multiitem multi-echelon problems, we demonstrate the effectiveness of a branch-and-cut algorithm using the proposed inequalities.