Dynamic Programming for the Minimum Tour Duration Problem

The minimum tour duration problem (MTDP) is the variant of the traveling salesman problem with time windows, which consists of finding a time window-feasible Hamiltonian path minimizing the tour duration. We present a new effective dynamic programming (DP)-based approach for the MTDP. When solving the traveling salesman problem with time windows with DP, two independent resources are propagated along partial paths, one for costs and one for earliest arrival times. For dealing with tour duration minimization, we provide a new DP formulation with three resources for which effective dominance and bounding procedures are applicable. This is a non-trivial task because in the MTDP at least two resources depend on each other in a non-additive and non-linear way. In particular, we define consistent resource extension functions (REF) so that dominance is straightforward using componentwise comparison for the respective resource vectors. Moreover, one of the main advantages of the new REF definition is that the DP can be reversed consistently such that the forward DP or any of its relaxations provides bounds for the backward DP, and vice versa. Computational test confirm the effectiveness of the proposed approach.