Exact solution of the soft-clustered vehicle-routing problem

The soft-clustered vehicle-routing problem (SoftCluVRP) extends the classical capacitated vehicle-routing problem by one additional constraint: The customers are partitioned into clusters and feasible routes must respect the soft-cluster constraint, that is, all customers of the same cluster must be served by the same vehicle. In this article, we design and analyze different branch-and-price algorithms for the exact solution of the SoftCluVRP. The algorithms differ in the way the column-generation subproblem, a variant of the shortest-path problem with resource constraints (SPPRC), is solved. The standard approach for SPPRCs is based on dynamic-programming labeling algorithms. We show that even with all the recent acceleration techniques (e.g., partial pricing, bidirectional labeling, decremental state space relaxation) available for SPPRC labeling algorithms, the solution of the subproblem remains extremely difficult. The main contribution is the modeling and solution of the subproblem using a branch-and-cut algorithm. The conducted computational experiments prove that branch-and-price equipped with this integer programming-based approach outperforms sophisticated labeling-based algorithms by one order of magnitude. The largest SoftCluVRP instances solved to optimality have more than 400 customers or more than 50 clusters.