A branch-and-price-and-cut algorithm for time-dependent pollution routing problem

The time-dependent pollution routing problem (TDPRP) extends the pollution routing problem (PRP) cause it captures traffic congestion at peak periods in urban transportation. It concerns planning a fleet of homogeneous vehicles to serve all customers, jointly deciding their speed on every arc and the departure time from each node such that the total cost of routes reaches a minimum. To our knowledge, all solution methods for the TDPRP are founded on (meta-)heuristics. In this study, we propose an exact branch-and-price-and-cut (BPC) algorithm for the TDPRP. The master problem is a route selection problem formulated as a set-partitioning model, and the pricing problem is a time-dependent elementary shortest path problem with resource constraints (TDESPPRC), in which the speed and start time at each customer need to be decided. We devise a tailored label-setting algorithm to handle the pricing problem, and the master problem is solved by column generation. Then, two valid inequalities are employed to tighten the lower bound, and several accelerating techniques are used to speed up the label-setting algorithm. Meanwhile, we present an analytical characterization applied to different scenarios. Finally, extensive computational experiments show our algorithm outperform a commercial MIP solver, and the optimal solutions are found for more benchmark instances using less time.