Scalable Timing-Aware Network Design via Lagrangian Decomposition

This paper addresses instances of the temporal fixed-charge multi-commodity flow (tfMCF) problem that arise in a very large scale dynamic transportation application. We model the tfMCF as a discrete-time Resource Task Network (RTN) with cyclic schedule, and formulate it as a mixed-integer program. These problems are notoriously hard to solve due to their time-expanded nature, and their size renders their direct solution difficult. We exploit synergies between flows of certain commodities in the formulation to devise model condensation techniques that reduce the number of variables and constraints by a factor of 25%-50%. We propose a solution algorithm that includes balanced graph partitioning, Lagrangian decomposition and a linear programming filtering heuristic. Computational results show that the proposed algorithm allows the solution of previously intractable instances, and the primal solution obtained by the heuristic step is within 2% duality gap of the linear relaxation of the original problem.