  As an effective model corresponding to $Z_3$-symmetric QCD ($Z_3$-QCD), we
construct a $Z_3$-symmetric effective Polyakov-line model ($Z_3$-EPLM) by using
the logarithmic fermion effective action. Since $Z_3$-QCD tends to QCD in the
zero temperature limit, $Z_3$-EPLM also agrees with the ordinary effective
Polyakov-line model (EPLM) there; note that ordinary EPLM does not possess
$Z_3$ symmetry. Our main purpose is to discuss a sign problem appearing in
$Z_3$-EPLM. The action of $Z_3$-EPLM is real, when the Polyakov line is not
only real but also its $Z_3$ images. This suggests that the sign problem
becomes milder in $Z_3$-EPLM than in EPLM. In order to confirm this suggestion,
we do lattice simulations for both EPLM and $Z_3$-EPLM by using the reweighting
method with the phase quenched approximation. In the low-temperature region,
the sign problem is milder in $Z_3$-EPLM than in EPLM. We also propose a new
reweighting method. This makes the sign problem very weak in $Z_3$-EPLM.
