Modeling and Solving the Vehicle Routing Problem with Stochastic Travel Times and Hard Time Windows

In practically all real-world planning and scheduling problems one is faced with uncertainty. Usually this uncertainty is abstracted away during the modeling process, such that standard combinatorial optimization algorithms can be used to solve the problem. This abstraction step is necessary, because these algorithms generally cannot deal with uncertainty. In this thesis we study the Vehicle Routing Problem with Stochastic Travel Times and Hard Time Windows (SVRP-HTW), a vehicle routing problem where we focus on uncertainty related to travel times. If the travel times are stochastic and their distribution is known, then they can be modeled using random variables. For solving the SVRP-HTW, an addition and a maximum operation for random variables is required. Therefore we developed a probabilistic arithmetic, which can be used for computing with arbitrary and independent random variables. It is sufficiently generic that it also can be used for solving other stochastic optimization problems. We show that this method is reasonably fast and is an improvement over Monte-Carlo simulation. As no benchmark for the SVRP-HTW was available, we created one based on the Solomon benchmark. We solved the instances using an existing implementation of a VRPTW solver, which we adapted, using our probabilistic arithmetic, for solving the SVRP-HTW. Our results show that our probabilistic arithmetic is sufficiently accurate and that its use does not degrade the effectiveness of the original VRPTW solver. However, it does significantly increase computation time.