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

General methods for synchromodal planning of freight containers and transports

Synchromodal freight transport is introduced as intermodal transport, so container transport that uses several transportation vehicles, with an increased focus on a-modal booking, cooperation and real-time flexibility.It is confirmed that general synchromodal network planning methods are rare or non-existent at the operational level. An extensive framework is developed that describes characteristics of different mathematical synchromodal optimisation problems on the tactical-operational levels.Three different problems are defined using this framework. Solution methods for these three problems are developed in this thesis, with a focus on low computation times so as to facilitate decision-support and real-time flexibility, and a focus on generality so as to make the methods applicable throughout different organisation structures.In the first problem, it is assumed that the transportation vehicles have fixed time tables and one only has to decide on a container-to-mode assignment, so by what modality-paths all containers reach their destination against minimal total cost. The containers have release times and deadlines. A model that also allows soft due dates is developed. Moreover, the option of using trucks or other ‘infinite resources’ to help fulfil requests is added. With appropriate graph reductions, this problem can be solved to optimality in little time by solvingthe minimum cost multi-commodity flow problem on an appropriate space-time network.In the second problem, the goal is the same but almost any element can be stochastic: for instance, travel times and container release times could be given a discrete probability distribution rather than a fixed value. Rigorous definitions are formulated to capture the generalities in this stochasticity. Multistage stochastic optimisation and Markov Decision Processes are illustrated, but advised against for their computing time: instead, Expected Future Iterationand 70%-Pessimistic Future Iteration are developed and shown to yield near-optimal results in a small amount of time in the simulated environment.In the final problem, there are no stochastic elements, but the decision-maker is given control over the vehicle time tables in addition to the control over container-to-mode assignments. This problem is argued to be a departure from classical optimisation problems, but shown to still be strongly NP-hard. An integer linear program is developed to solve the problem, butthe results show that it scales too poorly to solve problems of ‘real life size’ in an appropriate amount of time for decision support. The Greedy Gain heuristic and Compatibility Clustering heuristic are developed: they solve much more limited sub-problems with the ILP, but unfortunately, even these sub-problems require too much computational effort at the wished instance size.A number of topics for future research are formulated, giving concrete advice on how to solve the second problem more robustly and how to solve the third problem more quickly.Applied Mathematic

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.

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