Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs

Network flow problems with non-convex piecewise linear cost structures arise in many application areas, most notably in freight transportation and supply chain management. In the present paper, we consider mixed-integer programming (MIP) formulations of a generic multi-commodity network flow problem with piecewise linear costs. The formulations we study are based on variable disaggregation techniques, which have been used for a while to derive strong MIP formulations for variants of the fixed-charge network flow problem, a special case of our generic problem. To the best of our knowledge, variable disaggregation techniques have not been studied extensively within the framework of a general non-convex piecewise linear cost function, although a few authors have used them in this context (a recent example is the paper by Croxton, Gendron and Magnanti [2]). Given a directed network G = (V, A), with V , the set of nodes, A, the set of arcs, supplies and demands of multiple commodities at the nodes and capacities at the arcs, we consider the problem of finding the minimum cost multi-commodity flow when the objective is the sum of |A| piecewise linear functions. More specifically, on each arc a of the network, the cost is a function, ga, of the total flow, xa, on the arc, with the unit flow cost and fixed charge varying according to the flow on the arc. The function need not be continuous; it can have positive or negative jumps, though we do assume that the function is lower semi-continuous, that is, ga(xa) ≤ lim inf x a →xa ga(x ′ a) for