We consider a multiple-vehicle routing problem with split pick-ups (mVRPSP). This problem involves multiple suppliers, a single depot, and a fleet of identical capacity trucks responsible for delivering supplies from the suppliers to the depot. Any supplier may be visited by more than one truck, thus allowing split pick-ups. The problem is to determine, for each truck, which suppliers to visit and the size of loads to pick up so as to minimize the total transportation cost for the fleet, which depends on the number of trucks used and their routes. We develop a fundamentally new model for the mVRPSP, a deterministic dynamic program (DP). Although the most natural DP formulation results in a DP with uncountably-infinite state and action spaces, an optimality-invariance condition we establish leads to an equivalent DP with finite state and action spaces. This DP formulation leads to a new exact algorithm for solving the mVRPSP, based on a shortest path search algorithm, which is conceptually simple and easy to implement.