Capacitated production and subcontracting in a serial supply chain

This paper extends a series of recent results regarding the polynomial solvability of single and two-echelon deterministic lot sizing models. In particular, we consider a two-echelon supply chain in which production, inventory, transportation, backlogging, and subcontracting decisions are integrated. We allow for arbitrary concave cost functions and for stationary production and subcontracting capacities. We develop dynamic programming algorithms for various problems in this class that run in polynomial time in the planning horizon of the problem.