Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations

We consider the problem of routing vehicles stationed at a central facility (depot) to supply customers with known demands, in such a way as to minimize the total distance travelled. The problem is referred to as the vehicle routing problem (VRP) and is a generalization of the multiple travelling salesman problem that has many practical applications. We present tree search algorithms for the exact solution of the VRP incorporating lower bounds computed from (i) shortest spanning k-degree centre tree (k-DCT), and (ii) q-routes. The final algorithms also include problem reduction and dominance tests. Computational results are presented for a number of problems derived from the literature. The results show that the bounds derived from the q-routes are superior to those from k-DCT and that VRPs of up to about 25 customers can be solved exactly.