Approximation algorithms for k-echelon extensions of the one warehouse multi-retailer problem

In this paper, we consider k-echelon extensions of the deterministic one warehouse multi-retailer problem. We give constant factor approximation algorithms for some of these extensions when k is fixed. We focus first on the case without backorders and we give a $$(2k-1)$$ ( 2 k - 1 ) -approximation algorithm under general assumptions on the evolution of the holding costs as products move toward the final customers. We then improve this result to a k-approximation when the holding costs are monotonically non-increasing or non-decreasing (which is a natural situation in practice). Finally we address problems with backorders: we give a 3-approximation for the one-warehouse multi-retailer problem with backlog and a k-approximation algorithm for the k-level Joint Replenishment Problem with backlog (a variant where inventory can only be kept at the final retailers). Ours results are the first constant approximation algorithms for those problems. In addition, we demonstrate the potential of our approach on a practical case. Our preliminary experiments show that the average optimality gap is around 15%.