eCommerce, postal and logistics’ planners require to solve large-scale capacitated vehicle routing problems (CVRPs) on a daily basis. CVRP problems are NP-Hard and cannot be easily solved for large problem instances. Given their complexity, we propose a methodology to reduce the size of CVRP problems that can be later solved with state-of-the-art optimization solvers. Our method is an efficient version of clustering that considers the constraints of the original problem to transform it into a more tractable version. We call this approach Constrained Clustering Capacitated Vehicle Routing Solver (CC-CVRS) because it produces a soft-clustered vehicle routing problem with reduced decision variables. We demonstrate how this method reduces the computational complexity associated with the solution of the original CVRP and how the computed solution can be transformed back into the original space. Extensive numerical experiments show that our method allows to solve very large CVRP instances within seconds with optimality gaps of less than 16%. Therefore, our method has the following benefits: it can compute improved solutions with small optimality gaps in near real-time, and it can be used as a warm-up solver to compute an improved solution that can be used as an initial solution guess by an exact solver.

Application of deep learning to NP-hard combinatorial optimization problems is an emerging research trend, and a number of interesting approaches have been published over the last few years. In this work we address robust optimization, which is a more complex variant where a max-min problem is to be solved. We obtain robust solutions by solving the inner minimization problem exactly and apply Reinforcement Learning to learn a heuristic for the outer problem. The minimization term in the inner objective represents an obstacle to existing RL-based approaches, as its value depends on the full solution in a non-linear manner and cannot be evaluated for partial solutions constructed by the agent over the course of each episode. We overcome this obstacle by defining the reward in terms of the one-step advantage over a baseline policy whose role can be played by any fast heuristic for the given problem. The agent is trained to maximize the total advantage, which, as we show, is equivalent to the original objective. We validate our approach by solving min-max versions of standard benchmarks for the Capacitated Vehicle Routing and the Traveling Salesperson Problem, where our agents obtain near-optimal solutions and improve upon the baselines.

This study introduces a model to solve a dynamic network optimization model on a heterogeneous graph. We use this model to optimize the collection and consolidation operations on a cross-country multi-modal distribution network. The model's dynamic objects are trucks, trailers, orders, unvisited collection and customs check points. Information about dynamic objects is extracted from a realtime database. The model's static objects include objects that are known in advance, such as warehouses. The constraints of the problem include due dates, vehicle capacity, availability of vehicles, and precedence constraints of visiting locations. We propose a mixed-integer programming model and provide a solution using matheuristics. We decompose the master MIP model into subproblems that can be solved to optimality with LP solvers. We also reduce the graph complexity by variable fixing due to optimized subproblems or by bounding the maximum number of paths to be selected due to the solutions of priority-based bin packing algorithms. Finally, we convert the resulting problem into a bipartite matching problem by expanding the graph nodes which can then be solved in polynomial time. We implement our solution method on real-time data retrieved from the tracking system of a third-party logistics company. Experiments show that our solution method significantly outperforms other heuristics in terms of solution quality which is measured with respect to lateness, empty kilometers traveled, travel times, number of required/used vehicles, load factors, and ratio of served orders.