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In this paper, we introduce a branch-and-cut-and-price algorithm for the two-echelon capacitated vehicle routing problem. The algorithm relies on a reformulation based on q-routes that combines two important features. First, it overcomes symmetry issues observed in a formulation coming from a previous study of the problem. Second, it is strengthened with several classes of valid inequalities. As a result, the branch-and-cut-and-price implementation compares favorably with previous exact solution approaches for the problem—namely, two branch-and-price algorithms and a branch-and-cut method. Overall, 10 new optimality certificates and 8 new best upper bounds are provided in this study. New best lower bounds are also presented for all instances in the hardest test set from the literature.

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Fuel cost contributes to a significant portion of operating cost in cargo transportation. Though classic routing models usually treat fuel cost as input data, fuel consumption heavily depends on the travel speed, which has led to the study of optimizing speeds over a given fixed route. In this paper, we propose a joint routing and speed optimization problem to minimize the total cost, which includes the fuel consumption cost. The only assumption made on the dependence between the fuel cost and travel speed is that it is a strictly convex differentiable function. This problem is very challenging, with medium-sized instances already difficult for a general mixed-integer convex optimization solver. We propose a novel set partitioning formulation and a branch-cut-and-price algorithm to solve this problem. Our algorithm clearly outperforms the off-the-shelf optimization solver, and is able to solve some benchmark instances to optimality for the first time.