Alternative Formulations and Improved Bounds for the Multi-Depot Fleet Size and Mix Vehicle Routing

In this paper we model and solve the multi-depot fleet size and mix vehicle routing problem (MDFSMVRP). This problem extends the multi-depot vehicle routing problem and the fleet size and mix vehicle routing problem, two logistics problems that have been extensively studied for many decades. This difficult transportation problem combines complex assignment and routing decisions under the objective of minimizing fixed vehicle costs and variable routing costs. We first propose five distinct formulations to model the MDFSMVRP. We introduce a three-index formulation with an explicit vehicle index and a compact two-index formulation in which only vehicle types are identified. Other formulations are obtained by defining aggregated and disaggregated loading variables. The last formulation makes use of capacity-indexed variables. For each formulation, we propose sets of known and new valid inequalities to strengthen them, including symmetry breaking, lexicographic ordering, routing and rounded capacity cuts, among others. We then implement branch-and-cut and branch-and bound algorithms for these formulations. We compare the bounds provided by the formulations on a commonly used set of instances in the MDFSMVRP literature, containing up to nine depots and 360 customers, and on newly generated instances. Our in-depth analysis of the five formulations shows which formulations tend to perform better on each type of instance. Moreover, we have considerably improved the lower bounds on all instances and significantly improved the quality of the upper bounds that can be obtained by means of currently available exact methods. Acknowledgements. (NSERC). This support is gratefully acknowledged. We also thank Calcul Quebec for providing computing facilities.