A Maximum Expected Covering Problem for District Design

The optimal location of ambulances in a geographic region is interrelated with how the ambulances are dispatched to patients. Papers in the literature often treat the location and dispatching of ambulance separately. In this paper, we propose a novel mixed integer linear programming (MILP) model that determines how to locate and dispatch ambulances through district design. The model allows for uncertainty in both ambulance travel times and ambulance availability, and it maximizes the coverage level, i.e., the fraction of high-priority calls that can be responded to within a fixed-time threshold. The proposed MILP model determines the stations where ambulances should be located and assigns each call location to the open ambulance stations according to a preference list. The preference list is a rank ordering of the ambulances to assign to patients at a call location, from the most to the least preferred. The preference lists partition the region into a series of response districts that depend on ambulance availability, and the model balances the workload among the servers and maintains contiguity in the first priority response districts. The underlying ambulance queuing dynamics introduce nonlinearities to the model. To maintain a linear model, we use a Hypercube approximation model to estimate several of the inputs, and we provide an iterative algorithm to update the input parameters and solve the resulting MILP model. A computational example illustrates the modeling paradigm and solution algorithm using a real-world example. The results suggest that the reduction in coverage to maintain contiguity and balanced workloads among the ambulances is small.