The Traveling k-Median Problem: Approximating optimal network coverage

We introduce the Traveling k-Median Problem (TkMP) as a natural extension of the k-Median Problem, where k agents (medians) can move through a graph of n nodes over a discrete time horizon of ω steps. The agents start and end at designated nodes, and in each step can hop to an adjacent node to improve coverage. At each time step, we evaluate the coverage cost as the total connection cost of each node to its closest median. Our goal is to minimize the sum of the coverage costs over the entire time horizon. In this paper, we initiate the study of this problem by focusing on the uniform case, i.e., when all edge costs are uniform and all agents share the same start and end locations. We show that this problem is NP-hard in general and can be solved optimally in time O(ω2n2 k). We obtain a 5-approximation algorithm if the number of agents is large (i.e., k≥ n/ 2 ). The more challenging case emerges if the number of agents is small (i.e., k< n/ 2 ). Our main contribution is a novel rounding scheme that allows us to round an (approximate) solution to the ‘continuous movement’ relaxation of the problem to a discrete one (incurring a bounded loss). Using our scheme, we derive constant-factor approximation algorithms on path and cycle graphs. For general graphs, we use a different (more direct) approach and derive an O(min{ω,n}) -approximation algorithm if d(s,t)≤2ω, and an O(d(s,t)+ω) -approximation algorithm if d(s,t)>2ω, where d(s, t) is the distance between the start and end point

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

The enriched median routing problem and its usefulness in practice

Emergency response fleets often have to simultaneously perform two types of tasks: (1) urgent tasks requiring immediate action, and (2) non-urgent preventive maintenance tasks that can be scheduled upfront. In Huizing et al. (2020), Huizing et al. proposed the Median Routing Problem (MRP) to optimally schedule agents to a given set of non-urgent tasks, such that the response time for urgent tasks remains minimal. They proposed both an exact MILP-solution and a fast, scalable and accurate heuristic. However, when implementing the MRP-solution in a real-life pilot with a Dutch railway provider, we found that the model needed to be extended by including additional practical objectives and constraints. Therefore, in this paper, we extend the MRP to the so-called Enriched Median Routing Problem (E-MRP), making the model much better aligned with considerations from practice. Accordingly, we extend the MRP-based solutions to the E-MRP. This allows us to compare the performance of our proposed E-MRP solutions to performance obtained in the current operational practice of our partnering railway infrastructure company. We conclude that the E-MRP solution leads to a strong reduction in emergency response times compared to current practice by smartly scheduling the same volumes of non-urgent tasks

Optimising and recognising 2-stage delivery chains with time windows

In logistic delivery chains time windows are common. An arrival has to be in a certain time interval, at the expense of waiting time or penalties if the time limits are exceeded. This paper looks at the optimal placement of those time intervals in a specific case of a barge visiting two ports in sequence. For the second port a possible delay or penalty should be incorporated. Next, recognising these penalty structures in data is analysed to if see certain patterns in public travel data indicate that a certain dependency exists

Framework of synchromodal transportation problems

Though literature reviews of synchromodal transportation exist, no generalised mathematical model of these problems has been found yet. In this paper such a framework is introduced, by which mathematical models described in literature on synchromodal transportation problems can be classified. This framework should help researchers and developers to find solution methodologies that are commonly used in their problem instance and to grasp characteristics of the models and cases in a compact way, enabling easy classification, comparison and insight in complexity

Benchmarks and code underlying the publication: The Median Routing Problem for simultaneous planning of emergency response and non-emergency jobs

This is a collection of the 406 benchmark instances described in the linked publication, as well as a Python implementation of the described solution methods The goal of this entry is to ensure reproducibility of the results in the paper mentioned above. (Huizing, D., Schäfer, G., van der Mei, R., Bhulai, S. (2019). The Median Routing Problem for simultaneous planning of emergency response and non-emergency jobs.