A logic-based Benders decomposition approach for a fuel delivery problem with time windows, unsplit compartments, and split deliveries

We consider a single-period fuel delivery problem in which a distribution company has to transport multiple types of fuel from a depot to a set of service stations using a heterogeneous set of multi-compartment vehicles. Among other characteristics, the problem includes time windows, a limit on the duration of each route, unsplit compartments, and split deliveries. We propose two mixed integer programming (MIP) formulations and a logic-based Benders decomposition approach. The first formulation is an arc-based model while the second is based on the possible trips. The logic-based Benders decomposition follows a trip-based principle and breaks down the problem into a generalized assignment master problem and a subproblem responsible for identifying violated feasibility cuts implicated by the time-related constraints. It takes advantage of the problem-specific characteristics that allow efficiently solving the resulting subproblems. The logic-based Benders decomposition also serves as a heuristic, which works by limiting the number of trips generated throughout the process. Symmetry breaking constraints and preprocessing procedures are also proposed to help solving the formulations. The computational experiments using synthetic instances show that the logic-based Benders decomposition outperforms the other formulations and is very effective in solving the considered benchmark instances. It solved to optimality instances with up to 20 customers. The MIP heuristic obtained solutions within 7.2% of the optimal cost for all but one of the tested instances with up to 25 customers. Furthermore, it proved to be a viable approach for medium-sized instances where the exact logic-based Benders decomposition encountered difficulties.