To accommodate a swift response to fires and other incidents, fire departments have stations spread throughout their coverage area, and typically dispatch the closest fire truck(s) available whenever a new incident arises. However, it is not obvious that the policy of always dispatching the closest truck(s) minimizes the long-run fraction of late arrivals, since it may leave gaps in the coverage for future incidents. Although the research literature on dispatching of emergency vehicles is substantial, the setting with multiple trucks has received little attention. This is despite the fact that here careful dispatching is even more important, since the potential coverage gap is much larger compared to the single-truck case. Moreover, when dispatching multiple trucks, the uncertainty in the trucks' driving time plays an important role, in particular due to possible correlation in driving times of the trucks if their routes overlap.In this paper we discuss optimal dispatching of fire trucks, based on a particular dispatching problem that arises at the Amsterdam Fire Department, where two fire trucks are send to the same incident location for a quick response. We formulate the dispatching problem as a Markov Decision Process, and numerically obtain the optimal dispatching decisions using policy iteration. We show that the fraction of late arrivals can be significantly reduced by deviating from current practice of dispatching the closest available trucks, with a relative improvement of on average about 20%, and over 50% for certain instances. We also show that driving-time correlation has a non-negligible impact on decision making, and if ignored may lead to performance decrease of over 20% in certain cases. As the optimal policy cannot be computed for problems of realistic size due to the computational complexity of the policy iteration algorithm, we propose a dispatching heuristic based on a queueing approximation for the state of the network. We show that the performance of this heuristic is close to the optimal policy, and requires significantly less computational effort.

The effectiveness of a fire department is largely determined by its ability to respond to incidents in a timely manner. To do so, fire departments typically have fire stations spread evenly across the region, and dispatch the closest truck(s) whenever a new incident occurs. However, large gaps in coverage may arise in the case of a major incident that requires many nearby fire trucks over a long period of time, substantially increasing response times for emergencies that occur subsequently. We propose a heuristic for relocating idle trucks during a major incident in order to retain good coverage. This is done by solving a mathematical program that takes into account the location of the available fire trucks and the historic spatial distribution of incidents. This heuristic allows the user to balance the coverage and the number of truck movements. Using extensive simulation experiments we test the heuristic for the operations of the Fire Department of Amsterdam-Amstelland, and compare it against three other benchmark strategies in a simulation fitted using 10 years of historical data. We demonstrate substantial improvement over the current relocation policy, and show that not relocating during major incidents may lead to a significant decrease in performance.

In this paper we introduce a new model where the concept of condition-based maintenance is combined in a network setting with dynamic spare parts management.The model facilitates both preventive and corrective maintenance of geographically distributed capital goods as well as relocation of spare parts between different warehouses based on the availability of stock and the condition of all capital good installations. We formulate the problem as a Markov decision process, with the degradation process explicitly incorporated into the model. Numerical experiments show that that significant cost savings can be achieved when condition monitoring is used for preventive maintenance in a service network for capital goods.