Evaluating metaheuristic solution quality for a hierarchical vehicle routing problem by strong lower bounding

We study a vehicle routing problem that originates from a Nordic distribution company and includes the essential decision-making components of the company's logistics operations. The problem considers customer deliveries from a depot using heavy depot vehicles, swap bodies, optional switch points, and lighter local vehicles; a feature is that deliveries are made by both depot and local vehicles. The problem has earlier been solved by a fast metaheuristic, which does however not give any quality guarantee. To assess the solution quality, two strong formulations of the problem based on the column generation approach are developed. In both of these the computational complexity is mitigated through an enumeration of the switch point options. The formulations are evaluated with respect to the quality of the linear programming lower bounds in relation to the bounds obtained from a compact formulation. The strong lower bounding quality enables a significant reduction of the optimality gap compared to the compact formulation. Further, the bounds verify the high quality of the metaheuristic solutions, and for several problem instances the optimality gap is even closed.