The Robust Traveling Salesman Problem with Time Windows Under Knapsack-Constrained Travel Time Uncertainty

Abstract

We study the traveling salesman problem with time windows (TSPTW) under travel time uncertainty—modeled by means of an uncertainty set including all travel time vectors of interest. We consider a knapsack-constrained uncertainty set stipulating a nominal and a peak travel time for each arc and an upper bound [Formula: see text] on the sum of all deviations from the nominal times. Viewing the difference between the peak time and its nominal value as the maximum delay possibly incurred when traversing the corresponding arc, the problem we consider is thus to find a tour that remains feasible for up to [Formula: see text] units of delay. This differs from previous studies on robust routing under travel time uncertainty, which have relied on cardinality-constrained sets and only allow for an upper bound on the number of arcs with peak travel time. We propose an exact algorithm based on column generation and dynamic programming that involves effective dominance rules and an extension of the [Formula: see text]-tour relaxation proposed in the literature for the classical TSPTW. The algorithm is able to solve the robust TSPTW under both knapsack- and cardinality-constrained travel time uncertainty. Extensive computational experiments show that the algorithm is successful on instances with up to 80 customers. In addition, we study the impact of the two uncertainty sets on the trade-off between service quality and cost exhibited by the resulting solutions.