Charlemagne's challenge: the periodic latency problem.

Latency problems are characterized by their focus on minimizing the waiting time for all clients. We study periodic latency problems, a non-trivial extension of standard latency problems. In a periodic latency problem each client has to be visited regularly: there is a server traveling at unit speed, and there is a set of n clients with given positions. The server must visit the clients over and over again, subject to the constraint that successive visits to client i are at most qi time units away from each other. We investigate two main problems. In problem PLPP the goal is to find a repeatable route for the server visiting as many clients as possible, without violating their qi's. In problem PLP the goal is to minimize the number of servers needed to serve all clients. In dependence on the topol- ogy of the underlying network, we derive polynomial-time algorithms or hardness results for these two problems. Our results draw sharp separation lines between easy and hard cases.Latency problem; Periodicity; Complexity;

Probabilistic bounds on the k−k-Traveling Salesman Problem and the Traveling Repairman Problem

The $k-$traveling salesman problem ($k$-TSP) seeks a tour of minimal length that visits a subset of $k\leq n$ points. The traveling repairman problem (TRP) seeks a complete tour with minimal latency. This paper provides constant-factor probabilistic approximations of both problems. We first show that the optimal length of the $k$-TSP path grows at a rate of $\Theta\left(k/n^{\frac{1}{2}\left(1+\frac{1}{k-1}\right)}\right)$. The proof provides a constant-factor approximation scheme, which solves a TSP in a high-concentration zone -- leveraging large deviations of local concentrations. Then, we show that the optimal TRP latency grows at a rate of $\Theta(n\sqrt n)$. This result extends the classical Beardwood-Halton-Hammersley theorem to the TRP. Again, the proof provides a constant-factor approximation scheme, which visits zones by decreasing order of probability density. We discuss practical implications of this result in the design of transportation and logistics systems. Finally, we propose dedicated notions of fairness -- randomized population-based fairness for the $k$-TSP and geographical fairness for the TRP -- and give algorithms to balance efficiency and fairness

A concise guide to existing and emerging vehicle routing problem variants

Vehicle routing problems have been the focus of extensive research over the past sixty years, driven by their economic importance and their theoretical interest. The diversity of applications has motivated the study of a myriad of problem variants with different attributes. In this article, we provide a concise overview of existing and emerging problem variants. Models are typically refined along three lines: considering more relevant objectives and performance metrics, integrating vehicle routing evaluations with other tactical decisions, and capturing fine-grained yet essential aspects of modern supply chains. We organize the main problem attributes within this structured framework. We discuss recent research directions and pinpoint current shortcomings, recent successes, and emerging challenges

An Overview of Drone Energy Consumption Factors and Models

At present, there is a growing demand for drones with diverse capabilities that can be used in both civilian and military applications, and this topic is receiving increasing attention. When it comes to drone operations, the amount of energy they consume is a determining factor in their ability to achieve their full potential. According to this, it appears that it is necessary to identify the factors affecting the energy consumption of the unmanned air vehicle (UAV) during the mission process, as well as examine the general factors that influence the consumption of energy. This chapter aims to provide an overview of the current state of research in the area of UAV energy consumption and provide general categorizations of factors affecting UAV's energy consumption as well as an investigation of different energy models

Variable Neighborhood Descent Matheuristic for the Drone Routing Problem with Beehives Sharing

In contemporary urban logistics, drones will become a preferred transportation mode for last-mile deliveries, as they have shown commercial potential and triple-bottom-line performance. Drones, in fact, address many challenges related to congestion and emissions and can streamline the last leg of the supply chain, while maintaining economic performance. Despite the common conviction that drones will reshape the future of deliveries, numerous hurdles prevent practical implementation of this futuristic vision. The sharing economy, referred to as a collaborative business model that foster sharing, exchanging and renting resources, could lead to operational improvements and enhance the cost control ability and the flexibility of companies using drones. For instance, the Amazon patent for drone beehives, which are fulfilment centers where drones can be restocked before flying out again for another delivery, could be established as a shared delivery systems where different freight carriers jointly deliver goods to customers. Only a few studies have addressed the problem of operating such facilities providing services to retail companies. In this paper, we formulate the problem as a deterministic location-routing model and derive its robust counterpart under the travel time uncertainty. To tackle the computational complexity of the model caused by the non-linear energy consumption rates in drone battery, we propose a tailored matheuristic combining variable neighborhood descent with a cut generation approach. The computational experiments show the efficiency of the solution approach especially compared to the Gurobi solver