New dynamic programming algorithms for the resource constrained elementary shortest path problem

The resource-constrained elementary shortest path problem arises as a pricing subproblem in branch-and-price algorithms for vehicle routing problems with additional constraints. We address the optimization of the resource-constrained elementary shortest path problem and we present and compare three methods. The first method is a well-known exact dynamic programming algorithm improved by new ideas, such as bi-directional search with resource-based bounding. The second method consists of a branch-and-bound algorithm, where lower bounds are computed by dynamic programming with state space relaxation; we show how bounded bi-directional search can be combined with state space relaxation and we present different branching strategies and their hybridization. The third method, called decremental state space relaxation, is a new one; exact dynamic programming and state space relaxation are two special cases of this new method. The experimental comparison of the three methods is definitely favourable to decremental state space relaxation. Computational results are given for different kinds of resources, arising from the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the vehicle routing problem with capacities