Urban last-mile delivery services with time windows and parcel lockers: Model formulation and branch-and-price algorithm

Parcel lockers contribute to reducing delivery time and costs, providing logistics companies with solutions to address the challenges of last-mile delivery services. In this paper, we propose a routing problem with parcel lockers that incorporates a compensation-oriented reward strategy. The problem assumes that logistics companies can influence customer preferences through reward incentives, thereby obtaining flexible delivery solutions. To formulate this problem, we proposed a mixed-integer linear programming model, followed by a set-partitioning model. To solve the latter model, we developed a branch-and-price algorithm, whose subproblem is a variant of the elementary shortest path problem with resource constraints. This subproblem requires making decisions about node selection, allocation, and sequencing. We also presented a tailored bidirectional labeling algorithm with new label extension and dominance rules to solve the subproblem. Finally, we generated two types of instances derived from benchmark instances and conducted extensive experiments based on the instances to evaluate the performance of the BP algorithm.