  In our latest studies, by introducing the novel order-preserving (OP)
criterion, we have successfully addressed the widely concerned issue of the
previously published mapped weighted essentially non-oscillatory (WENO) schemes
that it is rather difficult to achieve high resolutions on the premise of
removing spurious oscillations for long-run simulations of the hyperbolic
systems. In the present study, we extend the OP criterion to the WENO-Z-type
schemes as the forementioned issue has also been extensively observed
numerically for these schemes. Firstly, we innovatively present the concept of
the generalized mapped WENO schemes by rewriting the Z-type weights in a
uniform formula from the perspective of the mapping relation. Then, we
naturally introduce the OP criterion to improve the WENO-Z-type schemes, and
the resultant schemes are denoted as MOP-GMWENO-X. Finally, extensive numerical
experiments have been conducted to demonstrate the benefits of these new
schemes. We draw the conclusion that, the convergence propoties of the proposed
schemes are equivalent to the corresponding WENO-X schemes. The major benefit
of the new schemes is that they have the capacity to achieve high resolutions
and simultaneously remove spurious oscillations for long simulations. The new
schemes have the additional benefit that they can greatly decrease the
post-shock oscillations on solving 2D Euler problems with strong shock waves.
