Formulation and a two-phase matheuristic for the roaming salesman problem: Application to election logistics

In this paper we investigate a novel logistical problem. The goal is to determine daily tours for a traveling salesperson who collects rewards from activities in cities during a fixed campaign period. We refer to this problem as the Roaming Salesman Problem (RSP) motivated by real-world applications including election logistics, touristic trip planning and marketing campaigns. RSP can be characterized as a combination of the traditional Periodic TSP and the Prize-Collecting TSP with static arc costs and time-dependent node rewards. Commercial solvers are capable of solving small-size instances of the RSP to near optimality in a reasonable time. To tackle large-size instances we propose a two-phase matheuristic where the first phase deals with city selection while the second phase focuses on route generation. The latter capitalizes on an integer program to construct an optimal route among selected cities on a given day. The proposed matheuristic decomposes the RSP into as many subproblems as the number of campaign days. Computational results show that our approach provides near-optimal solutions in significantly shorter times compared to commercial solvers

Branch-and-Cut for the split delivery vehicle routing problem with time windows

The split delivery vehicle routing problem with time windows (SDVRPTW) is a notoriously hard combinatorial optimization problem. First, it is hard to find a useful compact mixed-integer programming (MIP) formulation for the SDVRPTW. Standard modeling approaches either suffer from inherent symmetries (mixed-integer programs with a vehicle index) or cannot exactly capture all aspects of feasibility. Because of the possibility to visit customers more than once, the standard mechanisms to propagate load and time along the routes fail. Second, the lack of useful formulations has rendered any direct MIP-based approach impossible. Up to now, the most effective exact algorithms for the SDVRPTW have been branch-and-price-and-cut approaches using path-based formulations. In this paper, we propose a new and tailored branch-and-cut algorithm to solve the SDVRPTW. It is based on a new, relaxed compact model, in which some integer solutions are infeasible for the SDVRPTW. We use known and introduce some new classes of valid inequalities to cut off such infeasible solutions. One new class is path-matching constraints that generalize infeasible-path constraints. However, even with the valid inequalities, some integer solutions to the new compact formulation remain to be tested for feasibility. For a given integer solution, we build a generally sparse subnetwork of the original instance. On this subnetwork, all time-window-feasible routes can be enumerated, and a path-based residual problem then solved to decide on the selection of routes, the delivery quantities, and thereby the overall feasibility. All infeasible solutions need to be cut off. For this reason, we derive some strengthened feasibility cuts exploiting the fact that solutions often decompose into clusters. Computational experiments show that the new approach is able to prove optimality for several previously unsolved instances from the literature

The Split Delivery Vehicle Routing Problem with Time Windows and Customer Inconvenience Constraints

In classical routing problems, each customer is visited exactly once. By contrast, when allowing split deliveries, customers may be served through multiple visits. This potentially results in substantial savings in travel costs. Even if split deliveries are beneficial to the transport company, several visits may be undesirable on the customer side: at each visit the customer has to interrupt his primary activities and handle the goods receipt. The contribution of the present paper consists in a thorough analysis of the possibilities and limitations of split delivery distribution strategies. To this end, we investigate two different types of measures for limiting customer inconvenience (a maximum number of visits and the temporal synchronization of deliveries) and evaluate the impact of these measures on carrier efficiency by means of different objective functions (comprising variable routing costs, costs related to route durations, fixed fleet costs). We consider the vehicle routing problem with time windows in which split deliveries are allowed (SDVRPTW) and define the corresponding generalization that takes into account customer inconvenience constraints (SDVRPTW-IC). We design an extended branch-and-cut algorithm to solve the SDVRPTW-IC and report on experimental results showing the impact of customer inconvenience constraints. We finally draw useful insights for logistics managers on the basis of the experimental analysis carried out