A single-productMfacilities in-series production-planning model is analysed in which known demands on the last facility must be satisfied. The model considers concave production costs and piecewise concave inventory costs in the introduction of production capacity constraints. Backlogging of unsatisfied demand is permitted. The structure of optimal production schedules is exploited and then used to develop an efficient shortest-path algorithm for the systems with parallel production (inventory) cost functions. A numerical example is presented