Formulations for a Location-Routing Problem with Simultaneous Pickup and Delivery

In this paper, we introduce a new variant of Location-Routing Problem (LRP), namely the LRP with simultaneous pickup and delivery (LRPSPD). The LRPSPD is defined as finding locations of the depots and designing vehicle routes in such a way that pickup and delivery demand of each customer must be performed with same vehicle and the overall cost is minimized. We propose two polynomial-size mixed integer linear programming formulations for the problem and a family of valid inequalities to strengthen the formulations. While the first formulation is a two-index node-based formulation, the second one is a two-index flowbased formulation. Furthermore, we show that several special cases of the LRPSPD can be modeled in a similar manner. Finally, we perform extensive computational experiments to numerically evaluate the strengths of the proposed formulations and to identify formulation that is efficient in finding optimal or near-optimal solutions.