| Summary: | The depot locations have a significant effect on the transportation cost in the multi-depot vehicle routing problem. A two-tier particle swarm optimization framework is proposed, in which an external particle swarm optimization and an internal particle swarm optimization are used to determine the optimal depot locations and the optimal multi-depot vehicle routing problem solution, respectively. In the internal particle swarm optimization, a novel particle encoding scheme is used to minimize the computational cost by concurrently allocating the customers to depots, assigning the customers to vehicles, and determining the optimal routing path for each vehicle. The quality of the solutions is enhanced through a designed mutation local search with savings scheme. To verify the effectiveness of the proposed scheme, six standard multi-depot vehicle routing problem instances are tested and compared. It is shown that the use of the external particle swarm optimization scheme to optimize the multi-depot locations reduces the average routing distance obtained by the internal particle swarm optimization by around 13.16% on average. Furthermore, for a real-world case, the proposed two-tier particle swarm optimization scheme reduces the total routing cost by around 18%. Restated, the proposed particle swarm optimization algorithm provides an effective and efficient tool for solving practical multi-depot vehicle routing problems. Notably, the proposed scheme can be used as a reference model for obtaining the optimal locations in a variety of scheduling problems.
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