Bounded Rank Optimization for Effective and Efficient Emergency Response

Effective placement of emergency response vehicles (such as ambulances, fire trucks, police cars) to deal with medical, fire or criminal activities can reduce the incident response time by few seconds, which in turn can potentially save a human life. Owing to its adoption in Emergency Medical Services (EMSs) worldwide, existing research on improving emergency response has focused on optimizing the objective of bounded time (i.e. number of incidents served in a fixed time). Due to the dependence of this objective on temporal uncertainty, optimizing the bounded time objective is challenging. In this paper, we propose a new objective referred to as the bounded rank (which is the number of incidents served by a base station whose rank is below a bounded rank value) that has nice theoretical properties and serves as an indirect substitute for the bounded time objective. To understand the theoretical properties of this new objective in the context of the spatio-temporal uncertainty associated with emergency incidents, we first provide a Poisson Point Process (PPP) model of the emergency response problem. We then formally define the bounded rank objective in the context of the model and demonstrate that the bounded rank metric is monotone submodular. Due to the monotone submodularity of the objective, we can propose a greedy approach that can provide an {em a priori} guarantee of 50% from optimal and a much tighter {em posteriori} guarantee. Practically and more importantly, we demonstrate that optimizing this bounded rank objective on simulators validated on real data (and not just on the abstract PPP model) provides better results than the best known approach for optimizing bounded time objective.