The robust vehicle routing problem with time windows: Solution by branch and price and cut

We study the vehicle routing problem with time windows under demand uncertainty. Such a problem arises in fuel delivery to gas stations, to farms, or to production plants. We suggest a robust formulation where uncertain demand has support defined by cardinality constrained sets. The model ensures the feasibility of each route when demand values on the route change in predefined set. The range is customer specific, does not assume any probability distribution of demand, and may be estimated using historical demand data. We develop a branch-and-price-and-cut algorithm where the pricing problem is a robust resource constrained shortest path problem. We extend the rounded capacity inequalities to the robust case and suggest a separation procedure that solves a robust bin packing problem. The bound is further strengthened using 2-path and subset row inequalities and the full algorithm is tested on an extension of the benchmark Solomon instances. The robust schedules are simulated and compared to schedules obtained by solving the deterministic problem under varying levels and distributions of uncertainty. Simulation results show that the robust model provides superior protection against demand uncertainty. Over all scenarios, only 0.57% of the robust solutions turn out to be infeasible compared to 41.62% of the deterministic ones. Such protection against infeasibility comes with an 11.22% increase in routing costs on average.