A two-commodity flow formulation for the capacitated truck-and-trailer routing problem

In the capacitated truck-and-trailer routing problem (CTTRP), a limited fleet of capacitated trucks and trailers is available at a depot to serve a set of customers, of which some are only accessible by the truck alone. In this paper, we propose a two-commodity flow formulation for the CTTRP, which uses two sets of flow variables that model the flow of goods carried by trucks pulling a trailer and by trucks alone, respectively. We describe some valid inequalities for strengthening the formulation and develop a branch-and-cut algorithm for solving it. In our computational experiments, we consider both the CTTRP and its special case in which a single capacitated truck pulling an uncapacitated trailer is available. We report results on instances derived from known benchmark sets featuring up to 40 customers. The results show that our branch-and-cut solves 30-customer instances with either a single vehicle or loose capacity constraints with very high success rate. Of the more tightly capacitated 30-customer instances, about half can be solved. We also report results on benchmark instances of the single TTRP with satellite depots with up to 50 customers. In addition, we find that the lower bounds provided by the two-commodity flow formulation are superior to those of a corresponding one-commodity flow formulation. Finally, we study the effect of forbidding transfers between trucks and trailers.