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Most state-of-the-art algorithms for the Vehicle Routing Problem, such as Branch-and-Price algorithms or meta heuristics, rely on a fast feasibility test for a given route. We devise the first approach to approximately check feasibility in the Stochastic Vehicle Routing Problem with time windows, where travel times are correlated and depend on the time of the day. Assuming jointly normally distributed travel times, we use a chance constraint approach to model feasibility, where two different application scenarios are considered, depending on whether missing a customer makes the rest of the route infeasible or not. The former case may arise, e.g., in drayage applications or in the pickup-and-delivery VRP. In addition, we present an adaptive sampling algorithm that is tailored for our setting and is much faster than standard sampling techniques. We use a case study for both scenarios, based on instances with realistic travel times, to illustrate that taking correlations and time dependencies into account significantly improves the quality of the solutions, i.e., the precision of the feasibility decision. In particular, the nonconsideration of correlations often leads to solutions containing infeasible routes.

We consider stochastic problems in which both the objective function and the feasible set are affected by uncertainty. We address these problems using a K-adaptability approach, in which K solutions for a given problem are computed before the uncertainty dissolves and afterwards the best of them can be chosen for the realized scenario. We analyze the complexity of the resulting problem from a theoretical viewpoint, showing that, even in case the deterministic problem can be solved in polynomial time, deciding if a feasible solution exists is $$\mathcal {NP}$$ NP -hard for discrete probability distributions. Besides that, we prove that an approximation factor for the underlying problem can be carried over to our problem. Finally, we present exact approaches including a branch-and-price algorithm. An extensive computational analysis compares the performances of the proposed algorithms on a large set of randomly generated instances.