A general variable neighborhood search for the traveling salesman problem with time windows under various objectives

The traveling salesman problem with time windows (TSPTW) has wide practical applications in transportation and scheduling operations. We study the TSPTW in the deterministic case as well as under travel time uncertainty. We consider the three classical TSPTW variants with the objectives of minimizing cost, completion time, and tour duration. In addition, a new TSPTW variant that maximizes the minimum slack of a tour, i.e., the smallest time buffer between the arrival time and the end of the time window over all nodes visited on the tour, is introduced. We further address a variant of the TSPTW in which the arc travel times are uncertain. This problem is modeled by means of an uncertainty set, and the goal is to determine a tour that remains feasible in the worst case within a budget stipulating the sum of all travel time deviations from their nominal values. To solve all targeted problem variants, we develop a two-phase general variable neighborhood search (GVNS) that applies an efficient move evaluation approach within the local search. Extensive numerical experiments on benchmark instances from the literature show that our GVNS finds high-quality solutions that are competitive with those obtained by the state-of-the-art heuristics for all problem variants.