Median Routing Problems:Integrated optimisation of network maintenance and emergency response

Abstract

This thesis studies the shift planning of emergency response organisations and how to include plannable, preventive tasks into the schedules of the responders, such that they remain well spread in case of emergencies. Both halves of this planning have individually received significant attention throughout the past decades: that is, much is known about either positioning responders to minimise emergency response times (e.g. the k-Median Problem) or about efficiently visiting plannable jobs with a fleet of agents (e.g. the Vehicle Routing Problem). However, when trying to do both simultaneously, the literature appears to still be in its infancy. Such a simultaneous planning seems promising: more preventive work can get done, and if tasks are chosen that are spread well, then the spread of agents can be better than if they are bunched together at the same response station. In Chapter 2, the Median Routing Problem is introduced as a model to compute how to visit plannable jobs in a graph in a fixed-length time horizon using a fleet of emergency response agents, while minimising the expected response time to the next emergency in the graph. Aside from an exponential-time algorithm that can compute the optimal solution, heuristics are developed that can be used in practice. Most notably, the MDSA-algorithm is found to be an effective heuristic: on benchmark instances, including case study data, it finds a solution in 2.4 seconds on average, which is on average only 3.2% away from optimal. Because it is NP-complete to decide for the Median Routing Problem whether a feasible solution exists, it makes no sense to develop a polynomial-time approximation algorithm for the general problem. Instead, in Chapter 3, a special version is studied in the form of the (Uniform) Travelling k-Median Problem. A constant-factor approximation is given for specific graph metrics: namely, those graphs on which we can solve the problem in polynomial time if the agents are allowed to move ‘continuously’ over the graph instead of ‘discretely’. Using a rounding theorem, it is shown that these ‘continuous solutions’ can be rounded back with bounded loss. For all other metrics, we provide a polynomial-time algorithm with an approximation factor that depends linearly on the graph diameter and sublinearly on the length of the time horizon. In Chapter 5, the Enriched Median Routing Problem is posed as an extension of the Median Routing Problem with fifteen additional features. In the same spirit of MDSA, a more flexible algorithm is developed that still decomposes some of the decisions into smaller problems. In order to have this simpler problem again act on a small decision space, a subroutine is developed in Chapter 4 that can take an input graph and ‘sparsify’ it into a graph with much fewer nodes and edges that can still reasonably ‘represent’ the original graph. In Chapter 4, not only is it demonstrated that this sparsification works well in a sample of practical Median Routing Problem use cases, but a proven upper bound is also given on how much quality loss can occur through sparsification, by introducing and using the Mixed Integer Jester Game framework for explicit worst-case computation. Finally, in the second half of Chapter 5, the Enriched Median Routing Problem and its MDSA-inspired heuristic are compared to the current practice of a case study organisation. It is concluded that, indeed, the results of this thesis can help the case study organisation to simultaneously perform 31.0% more plannable work and decrease emergency response times by 14.2%. A solver based on this heuristic has been taken into use, and the further implementation of this solver and other outlooks are discussed in Chapter 6