A compact model for the clustered orienteering problem

Background: The Clustered Orienteering Problem is an optimization problem faced in last-mile logistics. The aim is, given an available time window, to visit vertices and to collect as much profit as possible in the given time. The vertices to visit have to be selected among a set of service requests. In particular, the vertices belong to clusters, the profits are associated with clusters, and the price relative to a cluster is collected only if all the vertices of a cluster are visited. Any solving methods providing better solutions also imply a new step towards sustainable logistics since companies can rely on more efficient delivery patterns, which, in turn, are associated with an improved urban environment with benefits both to the population and the administration thanks to an optimized and controlled last-mile delivery flow. Methods: In this paper, we propose a constraint programming model for the problem, and we empirically evaluate the potential of the new model by solving it with out-of-the-box software. Results: The results indicate that, when compared to the exact methods currently available in the literature, the new approach proposed stands out. Moreover, when comparing the quality of the heuristic solutions retrieved by the new model with those found by tailored methods, a good performance can be observed. In more detail, many new best-known upper bounds for the cost of the optimal solutions are reported, and several instances are solved to optimality for the first time. Conclusions: The paper provides a new practical and easy-to-implement tool to effectively deal with an optimization problem commonly faced in last-mile logistics.