Reaching the elementary lower bound in the vehicle routing problem with time windows

In this article, we present a comparative study of several strategies that can be applied to achieve the so‐called elementary lower bound in vehicle routing problems, that is, the bound obtained when all positive‐valued variables in an optimal solution of the linear relaxation of the set‐partitioning formulation correspond to vehicle routes without cycles. This bound can be achieved by solving the resource‐constrained elementary shortest path problem–an N P ‐hard problem–as the pricing problem in a column generation algorithm, but several other strategies can be used to ultimately produce the same lower bound in less computational effort. State‐of‐the‐art algorithms for vehicle routing problems rely on the quality of this lower bound to either bound the size of the search tree in a branch‐and‐price algorithm or the complexity of an enumeration procedure used to limit the number of variables in the set‐partitioning model. We consider several strategies for imposing elementarity that involve ng‐paths, strong degree constraints, and decremental state‐space relaxation. We compare the performance of these strategies on some selected instances of the vehicle routing problem with time windows.