We investigate a challenging task in ambulatory care, namely minimizing delays in patient transport. In practice, a limited number of vehicles is available for non-emergency transport. Furthermore, the dispatcher rarely has access to complete information when establishing a transport plan for dispatching the vehicles. If additional transport is requested on demand, then schedules have to be updated, which can lead to long waiting times. We model the scheduling of patient transports as a vehicle routing problem with general time windows and solve it as a mixed--integer linear problem that is modified whenever additional transport information becomes available. We propose a modeling approach that is designed to determine fair and stable plans. Furthermore, we show that the model can easily be modified when transports need to satisfy additional requirements e.g. during the Covid-19 pandemic. To show the applicability and efficiency of our modeling approach, we conduct a numerical study using historical data from the region of Middle Franconia. The results reveal and show that, in general, mathematical optimization methods for mixed-integer linear programs can significantly decrease delays and have considerable potential for optimized patient transportation.