Matheuristics based on iterative linear programming and slope scaling for multicommodity capacitated fixed charge network design

We present new matheuristics for the multicommodity capacitated fixed-charge network design problem (MCND). The matheuristics are based on combining iterative linear programming (ILP) methods and slope scaling (SS) heuristics. Each iteration alternates between solving a linear program obtained by adding pseudo-cuts and a restricted mixed-integer programming (MIP) model. The SS heuristic is used as a warm start to a state-of-the-art generic method that solves the restricted MIP model. The resulting ILP/SS matheuristics are compared against state-of-the-art heuristics for the MCND on a set of large-scale difficult instances. The computational results show that the approach is competitive: when performed for a time limit of 1  hour, it finds more best solutions than any other heuristic, using comparable running times; when performed for a time limit of 5  hours, it identifies an optimal solution for each instance for which an optimal solution is known and it is able to find new best solutions for some very hard instances.