Exact approaches for the pickup and delivery problem with loading cost

• Introduced the pickup and delivery problem with loading cost to the literature.
• Formulated the problem into an arc-flow model and a set-partitioning model.
• Proposed a branch-and-cut and a branch-and-price to solve the problem.
• Conducted a detailed analysis on the proposed algorithms.

In this paper, we propose a branch-and-cut algorithm and a branch-and-price algorithm to solve the pickup and delivery problem with loading cost (PDPLC), which is a new problem derived from the classic pickup and delivery problem (PDP) by considering the loading cost in the objective function. Applications of the PDPLC arise in healthcare transportation where the objective function is customer-centric or service-based. In the branch-and-price algorithm, we devise an ad hoc label-setting algorithm to solve the pricing problem and employ the bounded bidirectional search strategy to accelerate the label-setting algorithm. The proposed algorithms were tested on a set of instances generated by a common data generator in the literature. The computational results showed that the branch-and-price algorithm outperformed the branch-and-cut algorithm by a large margin, and can solve instances with 40 requests to optimality in a reasonable time frame.