Modeling and Solving Vehicle Routing Problems with Many Available Vehicle Types

Abstract

Vehicle routing problems (VRPs) involving the selection of vehicles from a large set of vehicle types are hitherto not well studied in the literature. Such problems arise at Volvo Group Trucks Technology, that faces an immense set of possible vehicle configurations, of which an optimal set needs to be chosen for each specific combination of transport missions. Another property of real-world VRPs that is often neglected in the literature is that the fuel resources required to drive a vehicle along a route is highly dependent on the actual load of the vehicle.

We define the fleet size and mix VRP with many available vehicle types, called many-FSMVRP, and suggest an extended set-partitioning model of this computationally demanding combinatorial optimization problem. To solve the extended model, we have developed a method based on Benders decomposition, the subproblems of which are solved using column generation, and the column generation subproblems being solved using dynamic programming; the method is implemented with a so-called projection-of-routes procedure. The resulting method is compared with a column generation approach for the standard set-partitioning model. Our method for the extended model performs on par with column generation applied to the standard model for instances such that the two models are equivalent. In addition, the utility of the extended model for instances with very many available vehicle types is demonstrated. Our method is also shown to efficiently handle cases in which the costs are dependent on the load of the vehicle.

Computational tests on a set of extended standard test instances show that our method, based on Benders’ algorithm, is able to determine combinations of vehicles and routes that are optimal to a relaxation (w.r.t. the route decision variables) of the extended model. Our exact implementation of Benders’ algorithm appears, however, too slow when the number of customers grows. To improve its performance, we suggest that relaxed versions of the column generation subproblems are solved and that the set-partitioning model is replaced by a set-covering model.

Appendix: The Extended Test Instances many-FSMVRP5

The sets of vehicle types of the instances in CT12 are extended as follows: the fixed and variable costs are defined using the spline function in Matlab according to

fcostEXT = interp1(capacity, fcost, capacityEXT,

     ’spline’);

vcostEXT = interp1(capacity, vcost, capacityEXT,

     ’spline’);

Here, capacity denotes a vector with the capacities of the original vehicle types, fcost (vcost) denotes a vector with the fixed (variable) costs of the original vehicle types, capacityEXT denotes a vector with the capacities of the extended set of vehicle types, and fcostEXT (vcostEXT) denotes the resulting vector with the fixed (variable) costs of the corresponding extended set of vehicle types.

For each of the instances in many-FSMVRP5, a vehicle type is removed if no route in the set $\cup _{k\in \mathcal{K}}\mathcal{R}_{k}$ has a total customer demand that is equal to the capacity of this vehicle type (except for the smallest capacity). This removal of a vehicle type does not exclude any optimal solution from the feasible set, since any route assigned to such a vehicle can be assigned to a smaller vehicle at a lower cost. No vehicle type was, however, removed from any of the instances in CT12.