The median routing problem for simultaneous planning of emergency response and non-emergency jobs

This paper studies a setting in emergency logistics where emergency responders must also perform a set of known, non-emergency jobs in the network when there are no active emergencies going on. These jobs typically have a preventive function, and allow the responders to use their idle time much more productively than in the current standard. When an emergency occurs, the nearest responder must abandon whatever job he or she is doing and go to the emergency. This leads to the optimisation problem of timetabling jobs and moving responders over a discrete network such that the expected emergency response time remains minimal. Our model, the Median Routing Problem, addresses this complex problem by minimising the expected response time to the next emergency and allowing for re-solving after this. We describe a mixed-integer linear program and a number of increasingly refined heuristics for this problem. We created a large set of benchmark instances, both from real-life case study data and from a generator. On the real-life case study instances, the best performing heuristic finds on average a solution only 3.4% away from optimal in a few seconds. We propose an explanation for the success of this heuristic, with the most pivotal conclusion being the importance of solving the underlying p-Medians Problem.