A generalized variable neighborhood search heuristic for the robust traveling salesman problem with time windows

We consider the traveling salesman problem with time windows under travel time uncertainty, and we model it as a robust optimization problem. We consider two variants of the problem that are defined by two different uncertainty sets based on the budget of uncertainty polytope (Bertsimas and Sim 2004). The first one is a previously used cardinality-constrained set stipulating that each travel time can take value within a given interval defined by a nominal and a peak value, but assuming that in the worst case at most Γ of them can attain their peak value. The second one was first used in this context by Bartolini et al. (2020) and stipulates instead that in the worst case the sum of deviations of the actual travel time values from their nominal values cannot exceed an upper bound ∆. We develop a general variable neighborhood search (GVNS) heuristic that can solve both variants of the problem. The algorithm is inspired by the GVNS algorithm proposed by da Silva and Urrutia (2010) for the deterministic traveling salesman problem with time windows but is adapted to handle uncertain travel times and features a modified perturbation phase. We report the results of extensive computational experiments that assess the quality of the solutions found by our heuristic against the best known upper upper bounds found by the exact algorithm of Bartolini et al. (2020).