Echelon Stock Formulation of Arborescent Distribution Systems: An Application to the Wagner-Whitin Problem

An arborescent distribution system is a multi-level system in which each installation receives input from a unique immediate predecessor and supplies one or more immediate successors. In this paper, it is shown that a distribution system with an arborescent structure can also be modelled using an echelon stock concept where at any instant the total echelon holding cost is accumulated at the same rate as the total conventional holding cost. The computational efficiency of the echelon model is tested on the well-known Wagner-Whitin type dynamic inventory lot-sizing problem, which is an intractable combinatorial problem from both mixed-integer programming (MIP) and constraint programming (CP) standpoints. The computational experiments show that the echelon MIP formulation is computationally very efficient compared to the conventional one, whereas the echelon CP formulation remains intractable. A CP/LP hybrid yields a substantial improvement over the pure CP approach, solving all tested instances in a reasonable time.