The exact solutions of the Riemann problems for the two different
perturbed macroscopic production models are considered and constructed
respectively for all the possible cases. It is found that the asymptotic
limits of solutions to the Riemann problem for the first kind of
perturbed macroscopic production model do not coverage to those of the
pressureless gas dynamics model, because the delta shock wave in the
limiting situation has different propagation speed and strength from
those for the pressureless gas dynamics model. In order to remedy it,
the second kind of perturbed macroscopic production model is proposed,
whose asymptotic limits of Riemann solutions are identical with those of
the pressureless gas dynamics model.