A comparison of formulations for a three-level lot sizing and replenishment problem with a distribution structure

Abstract

We address a three-level lot sizing and replenishment problem with a distribution structure (3LSPD), which is an extension of the one-warehouse multi-retailer problem (OWMR). We consider one production plant that produces one type of item over a discrete and finite planning horizon. The items produced are used to replenish warehouses and then retailers using direct shipments. Each retailer is linked to a unique warehouse and there are no transfers between warehouses nor between retailers. We also assume that transportation is uncapacitated. However, we consider the possibility of imposing production capacity constraints at the production plant level. The objective is to minimize the sum of the fixed production and replenishment costs and of the variable inventory holding costs at all three levels. We compare 13 different MIP formulations to solve the problem. All of these formulations are adapted from existing MIP formulations found in the one-warehouse multi-retailer literature, but most formulations are new in the context of the 3LSPD. We run experiments on both balanced and unbalanced networks. Our results indicate that the multi-commodity formulation is well suited for uncapacititated instances and that the echelon stock reformulations are better for capacitated instances. They also show that the richer formulations are not necessarily the best ones and that the unbalanced instances are harder to solve.