A Bayesian Optimization Model for Locating Emergency Service Units

This paper presents a new approach to solving the optimal unit location problem in a stochastic emergency service system that takes into account state transitions and unit availabilities. The goal is to minimize the system-wide mean response time, which is formulated as a combinatorial optimization problem. We show that this problem is NP-hard and develop lower and upper bounds for the optimal solution using a special case of the classic p-median problem. To solve the problem, we develop a Bayesian optimization algorithm that we show always converges to the optimal solution with a sublinear regret rate. We evaluate our approach through numerical experiments and a constructed study using real data from the St. Paul, Minnesota emergency response system and show that our model consistently and quickly converges to the optimal solution. We show how the optimal unit locations change as call rates increase and find that solutions obtained using the deterministic p-median model deteriorate as the traffic intensity increases. We also show how our approach can be adapted to optimize other objective functions.