In this paper we address the Vehicle Routing Problem with Pickups and Deliveries (VRPPD), an extension of the classical Vehicle Routing Problem (VRP) where we consider precedences among customers, and develop an Integer Linear Programming (ILP) based local search procedure. We consider the capacitated one-to-one variant, where a particular precedence must be satisfied between pairs of origin-destination customers. We extend the scheme proposed in De Franceschi et al.  for the Distance-Constrained Capacitated VRP, which has been successfully applied to other variants of the VRP. Starting from an initial feasible solution, this scheme follows the destroy/repair paradigm where a set of vertices is removed from the routes and reinserted by solving heuristically an associated ILP formulation with an exponential number of variables, named Reallocation Model. In this research, we propose two formulations for the Reallocation Model when considering pickup and delivery constraints and compare their behavior within the framework in terms of the trade off between the quality of the solutions obtained and the computational effort required. Based on the computational experience, the proposed scheme shows good potential to be applied in practice to this kind of problems and is a good starting point to consider more complex versions of the VRPPD.