Analysis of the selective traveling salesman problem with time‑dependent profts

Abstract

We consider a generalization of the selective traveling salesman problem (STSP) in which the beneft of visiting a location changes over time. This new problem, called the selective travelling salesman problem with time-dependent profts (STSP-TDP), is defned on a graph with time-dependent profts associated with the vertices, and consists of determining a circuit of maximal total proft. In the STSP-TDP the tour length must not exceed a maximum value, and its starting and ending times must both lie within a prespecifed planning horizon. This problem arises in planning tourist itineraries, mailbox collection, military surveillance, and water sampling, where the traveler accumulates diferent profts upon visiting the locations throughout the day. We focus on analyzing several variants of the problem depending on the shape of the time-dependent proft function. If this function is not monotonic, it may be worth visiting a site more than once. We propose formulations for the single-visit case and for when multiple visits are allowed, in which case the problem reduces to an STSP, which is adapted to be solved as a longest path problem. These formulations are then solved for piecewise-linear proft functions using a general-purpose solver, and tested on several artifcially created instances and on four TSPLib instances involving up to 535 vertices. A detailed analysis of the problem and the solution is performed.