We study the ground states of the pieces' model in the Fermi-Dirac statistics in the thermodynamic limit. In other words, we consider the minimizing configurations of $ n $ interacting fermions in an interval $ \Lambda $ divided into pieces by a Poisson point process, when $ \frac{n}{\vert \Lambda\vert}\to \rho>0 $ as $ \vert \Lambda \vert \to \infty $. We notice that a decomposition into groups of pieces arises from the hypothesis of finite-range pairwise interaction. Under assumptions of convexity and non-degeneracy of the subsystems, we get an almost complete factorization of any ground state. This method applies at least for groups comprising one or two particles. It improves the expansion of the thermodynamic limit of the ground state energy per particle up to the error $ O(\rho^{2-\delta}) $, with $ 0<\delta<1 $. It also provides an approximate ground state for the pieces' model.