Modeling and solving a corporate vehicle-sharing problem combined with other modes of transport

We consider a car-sharing problem in a company during business hours. The employees, located at one or several offices, have to travel to one or more appointments each with a fixed location and fixed start and end times and return to one of the offices afterwards. Each employee trip can be carried out with one out of several alternative modes of transport. The considered modes of transport are a company car from the company car pool, walking, public transport, bike, and taxi. The aim is to assign modes of transport to employee trips such that the total costs of covering the trips is minimized. We first consider that the company is operating a shared fleet of a single type of vehicle and then that the fleet consists of different vehicle types. By relying on minimizing the savings when using a vehicle compared to the cheapest alternative available mode of transport (which is used if no vehicle is assigned to a trip), we do not need to model the alternative modes explicitly. For the case where the vehicle fleet consists of a single type of vehicle, we model the vehicle-sharing problem as a minimum-cost flow problem. Secondly, if multiple types of vehicles are available the problem can be formulated as a multi-commodity flow problem. Since very efficient solution methods are available for these formulations, they are applicable in daily operations. We provide a comprehensive computational study for both cases on instances based on demographic, spatial, and economic data of Vienna. We show that our formulations for the problem solve these instances in a few seconds, which makes them usable in an online booking system. In the analysis, we discuss different potential settings. We study different sizes and compositions of the shared fleet, restricted sets of modes of transport, and variations of the objective function.