An Optimal Control Theory for the Traveling Salesman Problem and Its Variants

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.Comment: 24 pages, 8 figure

The Vehicle Routing Problem with Service Level Constraints

We consider a vehicle routing problem which seeks to minimize cost subject to service level constraints on several groups of deliveries. This problem captures some essential challenges faced by a logistics provider which operates transportation services for a limited number of partners and should respect contractual obligations on service levels. The problem also generalizes several important classes of vehicle routing problems with profits. To solve it, we propose a compact mathematical formulation, a branch-and-price algorithm, and a hybrid genetic algorithm with population management, which relies on problem-tailored solution representation, crossover and local search operators, as well as an adaptive penalization mechanism establishing a good balance between service levels and costs. Our computational experiments show that the proposed heuristic returns very high-quality solutions for this difficult problem, matches all optimal solutions found for small and medium-scale benchmark instances, and improves upon existing algorithms for two important special cases: the vehicle routing problem with private fleet and common carrier, and the capacitated profitable tour problem. The branch-and-price algorithm also produces new optimal solutions for all three problems

A Two-Stage Approach for Routing Multiple Unmanned Aerial Vehicles with Stochastic Fuel Consumption

The past decade has seen a substantial increase in the use of small unmanned aerial vehicles (UAVs) in both civil and military applications. This article addresses an important aspect of refueling in the context of routing multiple small UAVs to complete a surveillance or data collection mission. Specifically, this article formulates a multiple-UAV routing problem with the refueling constraint of minimizing the overall fuel consumption for all of the vehicles as a two-stage stochastic optimization problem with uncertainty associated with the fuel consumption of each vehicle. The two-stage model allows for the application of sample average approximation (SAA). Although the SAA solution asymptotically converges to the optimal solution for the two-stage model, the SAA run time can be prohibitive for medium- and large-scale test instances. Hence, we develop a tabu-search-based heuristic that exploits the model structure while considering the uncertainty in fuel consumption. Extensive computational experiments corroborate the benefits of the two-stage model compared to a deterministic model and the effectiveness of the heuristic for obtaining high-quality solutions.Comment: 18 page

Tour recommendation for groups

Consider a group of people who are visiting a major touristic city, such as NY, Paris, or Rome. It is reasonable to assume that each member of the group has his or her own interests or preferences about places to visit, which in general may differ from those of other members. Still, people almost always want to hang out together and so the following question naturally arises: What is the best tour that the group could perform together in the city? This problem underpins several challenges, ranging from understanding people’s expected attitudes towards potential points of interest, to modeling and providing good and viable solutions. Formulating this problem is challenging because of multiple competing objectives. For example, making the entire group as happy as possible in general conflicts with the objective that no member becomes disappointed. In this paper, we address the algorithmic implications of the above problem, by providing various formulations that take into account the overall group as well as the individual satisfaction and the length of the tour. We then study the computational complexity of these formulations, we provide effective and efficient practical algorithms, and, finally, we evaluate them on datasets constructed from real city data

An evolutionary algorithm for online, resource constrained, multi-vehicle sensing mission planning

Mobile robotic platforms are an indispensable tool for various scientific and industrial applications. Robots are used to undertake missions whose execution is constrained by various factors, such as the allocated time or their remaining energy. Existing solutions for resource constrained multi-robot sensing mission planning provide optimal plans at a prohibitive computational complexity for online application [1],[2],[3]. A heuristic approach exists for an online, resource constrained sensing mission planning for a single vehicle [4]. This work proposes a Genetic Algorithm (GA) based heuristic for the Correlated Team Orienteering Problem (CTOP) that is used for planning sensing and monitoring missions for robotic teams that operate under resource constraints. The heuristic is compared against optimal Mixed Integer Quadratic Programming (MIQP) solutions. Results show that the quality of the heuristic solution is at the worst case equal to the 5% optimal solution. The heuristic solution proves to be at least 300 times more time efficient in the worst tested case. The GA heuristic execution required in the worst case less than a second making it suitable for online execution.Comment: 8 pages, 5 figures, accepted for publication in Robotics and Automation Letters (RA-L