There are many ways of modeling the Asymmetric Traveling Salesman Problem (ATSP) and the related Precedence Constrained ATSP (PCATSP). In this paper we present new formulations for the two problems that result from combining precedence variable based formulations with network flow based formulations. The motivation for this work is a property of the so-called GDDL inequalities (see Gouveia and Pesneau, 2006), the " disjoint sub-paths " property, that is explored to create formulations that combine two (or more) disjoint path network flow based formulations. Several sets of projected inequalities, in the space of the arc and precedence variables, and in the spirit of many inequalities presented in Gouveia and Pesneau (2006), are obtained by projecting these network flow based formulations. Computational results are given for the PCATSP and the ATSP to evaluate the quality of the new inequalities.