Optimal Unit Locations in Emergency Service Systems with Bayesian Optimization

We model the facility location problem in an emergency service system as an optimization problem in which the objective is to minimize the system-wide mean response time, which requires exponential complexity to solve. We show that this problem is NP-hard and develop lower and upper bounds for the optimal solution from a special case of the classical p-median problem. We propose a Bayesian optimization solution to this problem that includes searching within feasible trust regions and adaptive swapping strategies. We show that our algorithm always converges to a globally optimal solution with a regret bound guarantee. Our algorithm consistently outperforms the p-median solution in numerical experiments and quickly converges to the optimal solution. We also apply our method to solve the optimal ambulance location problem in St. Paul, Minnesota, using one year of real data. We show that our method converges to the optimal solution very quickly. Our method can be applied to solve the optimal unit locations in the emergency service systems of the largest cities.