Optimal algorithm for the demand routing problem in multicommodity flow distribution networks with diversification constraints and concave costs

Distribution problems are of high relevance within the supply chain system. In real life situations various different commodities may flow in the distribution process. Furthermore, the connection between production and demand centres makes use of complex mesh networks that can include diversification constraints to avoid overcharged paths. In addition, the consideration in certain situations of economies of scale gives rise to non-linear cost functions that make it difficult to deal with an optimal routing scheme. This problem is well represented by the multicommodity flow distribution networks with diversification constraints and concave costs (MFDCC) problem. Here we present an optimal algorithm based on the Kuhn-Tucker optimality conditions of the problem and capable of supplying optimal distribution routes in such complex networks. The algorithm follows an iterative procedure. Each iteration constructive solutions are checked with respect to the Kuhn-Tucker optimality conditions. Solutions consider a set of paths transporting all the demand allowed by its diversification constraint (saturated paths), a set of empty paths, and an indicator path transporting the remaining demand to satisfy the demand equation. The algorithm reduces the total cost in the network in a monotonic sequence to the optimum. The algorithm was tested in a trial library and the optimum was reached for all the instances. The algorithm showed a major dependency with respect to the number of nodes and arcs of the graph, as well as the density of arcs in the graph.