Projections of the capacitated network loading problem

Consider an undirected network and a set of commodities with specified demands between various pairs of nodes of the network. Given two types of capacitated facilities that can be installed (loaded) for arc dependent costs, we have to determine the integer number of facilities to load on each arc in order to send the required flow of all commodities at minimum total cost. We present a natural mixed-integer programming formulation of the problem and then consider its single commodity and multicommodity versions. We develop ‘equivalent’ formulations in a lower-dimensional space by projecting out the flow variables and study the polyhedral properties of the corresponding projection cones. Our results strengthen an existing result for multicommodity flow problems. We also characterize several classes of facet defining inequalities for this lower-dimensional polyhedron, and conclude by identifying some open problems and future research directions.