  The traditional Capacitated Vehicle Routing Problem (CVRP) minimizes the
total distance of the routes under the capacity constraints of the vehicles.
But more often, the objective involves multiple criteria including not only the
total distance of the tour but also other factors such as travel costs, travel
time, and fuel consumption.Moreover, in reality, there are numerous implicit
preferences ingrained in the minds of the route planners and the drivers.
Drivers, for instance, have familiarity with certain neighborhoods and
knowledge of the state of roads, and often consider the best places for rest
and lunch breaks. This knowledge is difficult to formulate and balance when
operational routing decisions have to be made. This motivates us to learn the
implicit preferences from past solutions and to incorporate these learned
preferences in the optimization process. These preferences are in the form of
arc probabilities, i.e., the more preferred a route is, the higher is the joint
probability. The novelty of this work is the use of a neural network model to
estimate the arc probabilities, which allows for additional features and
automatic parameter estimation. This first requires identifying suitable
features, neural architectures and loss functions, taking into account that
there is typically few data available. We investigate the difference with a
prior weighted Markov counting approach, and study the applicability of neural
networks in this setting.
