This paper studies the two-echelon capacitated vehicle routing problem with time windows. The first echelon consists of transferring freight from depots to intermediate facilities (i.e., satellites), whereas the second echelon consists of transferring freight from these facilities to the final customers, within their time windows. We propose two pathbased mathematical formulations for our problem: (1) in one formulation, paths are defined over both firstand second-echelon tours, and (2) in the other one, the firstand second-echelon paths are decomposed. Branch-and-price-based algorithms are developed for both formulations. We compare both formulations and solution methods on a comprehensive set of instances and are able to solve instances up to five satellites and 100 customers to optimality. This paper is the first paper in the literature that solves such large instance sizes.
Here we are dealing with minimum cost flow problem on dynamic network flows with zero transit times and a new arc capacity, horizon capacity, which denotes an upper bound on the total flow traversing through on an arc during a pre-specified time horizon T. We develop a simple approach based on mathematical modelling attributes to solve the min-cost dynamic network flow problem where arc capacities and costs are time varying, and horizon capacities are considered. The basis of the method is simple and relies on the appropriate defining of polyhedrons, and in contrast to the other usual algorithms that use the notion of time expanded network, this method runs directly on the original network.
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