A unified exact approach for a broad class of vehicle routing problems with simultaneous pickup and delivery

The vehicle routing problem (VRP) with simultaneous pickup and delivery (VRPSPD) is a classical combina-torial optimization problem in which one aims at determining least-cost routes, starting and ending at the depot, while simultaneously meeting the pickup and delivery demands of geographically dispersed customers in a single visit, and without violating the capacity of the vehicle. In this work, we propose a unified branch -cut-and-price (BCP) algorithm to solve ten variants of the problem, considering not only the standard version but also those including additional attributes such as a heterogeneous fleet of vehicles, time windows, route duration, multiple depots, and location decisions. The resulting problem that generalizes all of these variants can be formulated as a heterogeneous location routing problem with simultaneous pickup and delivery and time windows (HLRPSPDTW), for which we propose a compact formulation to serve as the basis for an extended formulation associated with the BCP approach. The proposed exact method includes many of the features developed in recent years, such as a bucket graph-based labeling algorithm, rank-1 cuts with limited memory, and dynamic ng-sets. Extensive computational experiments were performed on more than 550 benchmark instances and our algorithm was capable of obtaining a number of new optimal solutions, as well as improved lower bounds, for all variants addressed in this paper.