A multi-commodity flow formulation for the generalized pooling problem

The pooling problem is an extension of the minimum cost network flow problem where the composition of the flow depends on the sources from which it originates. At each source, the composition is known. In all other nodes, the proportion of any component is given as a weighted average of its proportions in entering flow streams. The weights in this average are simply the arc flow. At the terminals of the network, there are bounds on the relative content of the various components. Such problems have strong relevance in e.g. planning models for oil refining, and in gas transportation models with quality constraints at the reception side. Although the pooling problem has bilinear constraints, much progress in solving a class of instances to global optimality has recently been made. Most of the approaches are however restricted to networks where all directed paths have length at most three, which means that there is no connection between pools. In this work, we generalize one of the most successful formulations of the pooling problem, and propose a multi-commodity flow formulation that makes no assumptions on the network topology. We prove that our formulation has stronger linear relaxation than previously suggested formulations, and demonstrate experimentally that it enables faster computation of the global optimum.