An evaluation of common modeling choices for the vehicle routing problem with stochastic demands

Abstract

We investigate three common modeling choices for the vehicle routing problem with stochastic demands: (i) the total expected demand of customers on a route may not exceed the capacity of the vehicle, (ii) the number of routes is fixed, and (iii) demand is distributed with a support that contains negative-valued realizations. We prove that modeling choices (i) and (ii) result in an arbitrarily large increase of the optimal objective value in the worst case. Additionally, we provide lower and upper bounds on the change of the optimal objective value following from (iii) in case the actual distribution of demand is censored, truncated or folded. We also evaluate the consequences of these choices numerically, by employing a state-of-the-art integer $L$-shaped method to solve the vehicle routing problem with stochastic demands to optimality, which we modify to deal with the alternative choices. We find that restricting the expected demand of a route to the vehicle’s capacity has a limited effect on the optimal objective value for most, but not all, benchmark instances from the literature, while drastically reducing the computation times of the integer $L$-shaped method. When restricting the number of routes, a similar effect occurs when the total expected demand on a route is not restricted. Otherwise, the computation time decreases only slightly, and even increases for some benchmark instances. For instances from the literature, despite admitting negative realizations, the normal distributions used to model demand are an adequate approximation for censored, truncated and folded normal distributions that have nonnegative supports.