  In the article, we study the associated production of prompt $J/\psi$ mesons
and $Z(W)$ bosons in the improved color evaporation model using the high-energy
factorization as it is formulated in the parton Reggeization approach. The last
one is based on the modified Kimber-Martin-Ryskin-Watt model for unintegrated
parton distribution functions and the effective field theory of Reggezied
gluons and quarks, suggested by L.N.~Lipatov. We predict cross section for
associated $J/\psi$ and $Z ( W)$ hadroproduction via the single and double
parton scattering mechanisms using the set of model parameters that have been
obtained early for the description of single and double prompt $J/\psi$
production at the LHC energies. The numerical calculations are realized using
the Monte-Carlo event generator KaTie. The calculation results are compared
with the data at the energies $\sqrt{s} = 7 \mbox{ and } 8$ TeV and we make
predictions for the energy $\sqrt{s} = 13$ TeV for $J/\psi$ plus $Z(W)$ and
$\Upsilon$ plus $Z/W$ at the $\sqrt{s} = 8 \mbox{ and } 13$ TeV associated
productions.
