Production and distribution lot sizing in a two stage supply chain

We develop a two stage model of a manufacturing supply chain. This two stage production transportation model features capacitated production in two stages, and a fixed cost (or concave cost) for transporting the product between the stages. We prove several properties of this model, which we call the Two Stage Production Distribution Problem (2SPDP) model. By placing non-speculative assumptions on production and transportation, we show that our model reduces to a related model, with one capacitated production stage with linear production cost, and transportation between two inventory locations with non-linear transportation cost. Finally, we present polynomial algorithms for this model under several different transportation cost structures and capacity assumptions.