Lower Bounds and an Exact Method for the Capacitated Vehicle Routing Problem

In this paper we consider the problem in which a fleet of M vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. This problem is referred as the capacitated vehicle routing problem (CVRP). We present an exact algorithm for solving the CVRP based on a set partitioning formulation of the problem. We describe a procedure for computing a valid lower bound to the cost of the optimal CVRP solution that finds a feasible solution of the dual of the LP-relaxation of the set partitioning formulation without generating the entire set partitioning matrix. The dual solution obtained is then used to limit the set of the feasible routes containing the optimal CVRP solutions. The resulting set partitioning problem is solved by using a branch and bound algorithm. Computational results are presented for a number of problems derived from the literature. The results show the effectiveness of the proposed method in solving problems up to about 100 customers