A recursive simulation-optimization tool to estimate the busy fractions in the ambulance location and dispatching problem

Emergency Medical Services (EMS) are crucial in health care systems. They provide out-of-hospital acute medical care and transportation to an appropriate health center for injured and ill people. The Ambulance Location and Dispatching Problem (ALDP) identifies the location of the available ambulances and provides at the same time a dispatching policy, in order to minimize the expected response time to answer the calls. In particular, the ALDP considers an ordered dispatching list for each demand zone. When a call arises from a zone, the first available ambulance in its dispatching list is sent to answer the call; if no vehicles in the list are available, the closest available ambulance is sent; if no vehicles are available at all, the call is redirected to another service. However, the uncertain nature of the calls makes it impossible to know in advance if the ambulance identified by the dispatching policy will be available or not when the call arises. Thus, the probability that a vehicle is busy, denoted as busy fraction, is considered in the ALDP. While the description in terms of busy fractions is common in the literature, the corresponding probabilities that ambulances answer a call can be estimated in several manners. We propose in this work a new recursive simulationoptimization framework to fairly estimate them. In particular, we develop and compare four approaches, which are based on either the Bernoulli probability law or the queuing theory. Tests are conducted on realistic instances inspired by the city of Montréal, Québec, Canada. Results are evaluated in terms of the simulated response time and the gap between the planned response time from the optimization and the simulated one. Numerical results confirm that all approaches (especially the most complicated one) improve the performance with respect to an a priori evaluation of the busy fraction.