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An importer in Hong Kong employs vehicles, all from external transport companies, to deliver products to its customers geographically scattered in different locations. The delivery plan needs to simultaneously minimize the total traveling cost and balance the profits among all transport companies. This transportation practice engenders a new variant of vehicle routing problems, called the vehicle routing problem with outsourcing and profit balancing (VRPOPB). The profits are balanced by maximizing the minimum unit profit of all transport companies, which can effectively avoid the occurrence of distorted solutions. We develop two multi-objective local search (MOLS) algorithms for the problem, where the second one enhances the first one by incorporating several additional techniques. To evaluate our algorithms, we conduct extensive experiments on 57 generated instances and a real case obtained from a food importer in Hong Kong. The computational results clearly demonstrate that our enhanced MOLS algorithm is able to achieve satisfactory solutions.
Zhang et al. (2019) studied this question.