Nested column generation for split pickup vehicle routing problem with time windows and time-dependent demand

This study attempts to solve a split pickup vehicle routing problem with time windows and time-dependent demand. This is a vehicle routing problem in which demand is generated at different constant rates for different customers, and vehicles are allowed to pick up demand during the demand generation process. In this problem, the pickup time determines the quantity of demand that can be picked up, and a partial pickup is permitted during a single visit. The problem is formulated into a mixed-integer nonlinear programming model, and a sequence-extended network is designed to represent the relationship between multiple pickups. Because of the interdependences of time, load, and cost in the pricing subproblem, the classic labelling algorithm with weak dominance rules is ineffective, therefore, a nested column generation-based branch-and-price-and-cut algorithm is proposed to tackle the problem. The nested structure allows these trade-offs to be resolved in a lower-level column generation, and it can be applied to other interdependent routing problems. The computational results show that the proposed algorithm is effective in obtaining optimal solutions for small- and medium-scale instances within a reasonable time limit.