Scheduling electric vehicles

The vehicle scheduling problem (VSP) is a traditional problem in public transport. One of the main assumptions is that buses can be operated the whole day without any interruption for refueling etc. Recently, new technological innovations have led to the introduction of electric vehicles (EVs). For these new vehicles, we cannot ignore the need of refueling during the day, as the range of an electric bus is severely limited, because of the capacity of the batteries. In this paper, we study the electric VSP (e-VSP), where we use EVs with a limited range. During the day the batteries can be charged; in this paper we assume that a battery cannot be replaced/substituted. We present two models that differ in the level of detail resembling the actual processes. In our first model, we assume a linear charging process, work with a constant price of electricity during the day, and do not take the effect of the depth-of-discharge on the lifetime of the battery into account. Our second model resembles practice much better: we allow any type of charging process, work with the actual electricity prices, and take the depreciation cost of the battery into account. To keep this model tractable, however, we approximate the exact value of the charge by discretizing it. The refined model can be solved to optimality using integer linear programming for instances of small/medium size, and therefore, we describe two other solution methods based on column generation that find good, but not necessarily optimal, solutions for large instances. We have tested our algorithms on real-world instances.