We have developed a hybrid computer simulation method that can deal with an iso-thermal polymer melt spinning process at a steady state. In the hybrid melt spinning process simulation, we divide the simulation into two different spatial scales, macroscopic and microscopic level. At the macroscopic level, we solved macroscopic variables, such as local velocity and cross-sectional area of the fiber, based on a 1-d mathematical model developed by Kase and Matsuo, except for the stress appeared in the macroscopic equation. To evaluate the local stress of the fiber, we use a single Lagrange fluid element that can move on the spinning line according to the velocity evaluated at the macroscopic equation. The Lagrangian particle has a molecular dynamics simulation system and the microscopic system experiences a local deformation rate evaluated by the macroscopic flow at the place of the particle on the spinning line. The local stress at the place of the Lagrangian particle is evaluated by molecular dynamics simulation where the Kremer-Grest model is used as the model of the constituent polymer chains of the fiber. While exchanging the information between two levels, the hybrid simulations are repeated by the time the system reached a steady state. In this way, a steady state for a given draw ratio is obtained in a self-consistent manner. As results of the simulations mentioned above for various draw ratios, we could obtain not only the usual macroscopic information, such as local velocity, and cross-sectional area of the fiber, but also microscopic information of polymer chain at an arbitrary place on the spinning line, e.g., degree of stretching and orientation, spatial correlations of polymer chains.