In emergency response, cost minimization is usually not the target, but serving all incidents as well as possible, subject to a constraint on the budget or the number of locations. Also, serving ALL incidents well may be more important than serving incidents well on average.
In the p-center problem, we instead minimize the maximum distance between an incident and the assigned emergency response location. Modeling this min-max objective is not straight-forward, but with an additional constraint, we can also linearize this.
