We perform a theoretical study of the nonlinear optical response of an ultrathin film consisting of oriented linear aggregates. A single aggregate is described by a Frenkel exciton Hamiltonian with uncorrelated on-site disorder. The exciton wave functions and energies are found exactly by numerically diagonalizing the Hamiltonian. The principal restriction we impose is that only the optical transitions between the ground state and optically dominant states of the one-exciton manifold are considered, whereas transitions to other states, including those of higher exciton manifolds, are neglected. The optical dynamics of the system is treated within the framework of truncated optical Maxwell-Bloch equations, in which the electric polarization is calculated by using a joint distribution of the transition frequency and the transition dipole moment of the optically dominant states. This function contains all the statistical information about these two quantities that govern the optical response and is obtained numerically by sampling many disorder realizations. We derive a steady-state equation that establishes a relationship between the output and input intensities of the electric field and show that within a certain range of the parameter space this equation exhibits a three-valued solution for the output field. A time-domain analysis is employed to investigate the stability of different branches of the three-valued solutions and to get insight into switching times. We discuss the possibility to experimentally verify the bistable behavior.