Towards a connection between the capacitated vehicle routing problem and the constrained centroid-based clustering

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Efficiently solving a vehicle routing problem (\(\mathcal{VRP}\)) in a practical runtime is a critical challenge for delivery management companies. This paper explores both a theoretical and experimental connection between the Capacitated Vehicle Routing Problem (\(\mathcal{CVRP}\)) and the Constrained Centroid-Based Clustering (\(\mathcal{CCBC}\)). Reducing a \(\mathcal{CVRP}\) to a \(\mathcal{CCBC}\) is a synonym for a transition from an exponential to a polynomial complexity using commonly known algorithms for clustering, i.e K-means. At the beginning, we conduct an exploratory analysis to highlight the existence of such a relationship between the two problems through illustrative small-size examples and simultaneously deduce some mathematically-related formulations and properties. On a second level, the paper proposes a \(\mathcal{CCBC}\) based approach endowed with some enhancements. The proposed framework consists of three stages. At the first step, a constrained centroid-based clustering algorithm generates feasible clusters of customers. This methodology incorporates three enhancement tools to achieve near-optimal clusters, namely: a multi-start procedure for initial centroids, a customer assignment metric, and a self-adjustment mechanism for choosing the number of clusters. At the second step, a traveling salesman problem (\(\mathcal{TSP}\)) solver is used to optimize the order of customers within each cluster. Finally, we introduce a process relying on routes cutting and relinking procedure, which calls upon solving a linear and integer programming model to further improve the obtained routes. This step is inspired by the ruin & recreate algorithm. This approach is an extension of the classical cluster-first, route-second method and provides near-optimal solutions on well-known benchmark instances in terms of solution quality and computational runtime, offering a milestone in solving \(\mathcal{VRP}\).

, 26 pages

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