Vehicle Routing Problem with Simultaneous Pickup and Delivery: Mixed Integer Programming Formulations and Comparative Analyses

This paper addresses the vehicle routing problem with simultaneous pickup and delivery (VRPSPD) which is the problem of optimally integrating goods deliver and pickup when there are no precedence restrictions on the order in which the operations must be performed. We review mixed integer programming formulations in the literature for VRPSPD and propose two new formulations, PM1 and PM2, for the problem. While PM1 is a two-index node-based formulation, PM2 is a two-index flow-based formulation and both formulations have O(n) binary variables and O(n) constraints. Additionally, we prove that PM2 produce better lower bounds than PM1. To identify the efficient formulation, we conduct an experimental study and compare existing and proposed formulations in terms of linear programming relaxations gap, optimality gap, the number of optimal solutions obtained within two hours of CPU and average CPU time on two sets of test problems adapted from the literature. The computational results indicate that PM2 outperforms the other formulations in terms of the above performance measures.