Summary: We propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems. These formulations are obtained by lifting, using the reformulation-linearization technique, a novel polynomial size ordered path-based formulation. We show that the strongest proposed formulation is equivalent to the strongest multi-commodity flow formulation presented so far. In addition, we propose polynomial size valid inequalities that aim at further tightening the proposed formulations. Also, the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs. We show that the new derived model is not comparable with a state-of-the-art polynomial length formulation. We present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities.