Dynamic lot sizing for multi-echelon distribution systems with purchasing and transportation price discounts

The authors consider the problem of determining optimal purchasing and shippiing quantities over a finite planning horizon for arborescent, multi-echelon physical distribution systems with deterministic, time-varying demands. They assume that the inventory holding cost at a given warehouse of the distribution network is a linear function of the inventory level, and that the total procurement cost (i.e., ordering, plus purchasing, plus transportation and reception costs) is a general piecewise-linear function of the quantities shipped to and from the warehouse. The authors formulate a mixed integer linear programming model of the problem and develop a Lagrangian relaxation-based procedure to solve it. They show computational results for problems with 12 periods, up to 15 warehouses, and 3 transportation price ranges.