We study an exact model of radially self-similar magnetohydrodynamic outflows appropriate to describe collimated jets from astrophysical systems with accretion disks spiraling around a central gravitating object. The mathematical problem is reduced to the integration of a set of nonlinear ordinary differential equations. A physically accepted solution needs to cross several critical surfaces whose knowledge is not known a priori but depends on a number of physical parameters. We search for the right combination of those parameters which produces a superAlfvenic solution crossing the modified fast critical surface, satisfying thus causality. We use a numerical algorithm that tests pre-defined combinations of the problem parameters, classifies the derived solutions and tunes one parameter in each combination so that this solution crosses the modified fast critical surface. The results are discussed in the context of the launching and collimation properties of magnetohydrodynamic disk-winds and jets associated with young stellar objects in star formation regions.