  Quantum master equations are common tools to describe the dynamics of
many-body systems open to an environment. Due to the interaction with the
latter, even for the case of non-interacting electrons, the computational cost
to solve these equations increases exponentially with the particle number. We
propose a simple scheme, that allows to study the dynamics of $N$
non-interacting electrons taking into account both dissipation effects and
Fermi statistics, with a computational cost that scales linearly with $N$. Our
method is based on a mapping of the many-body system to a specific set of
effective single-particle systems. We provide detailed numerical results
showing excellent agreement between the effective single-particle scheme and
the exact many-body one, as obtained from studying the dynamics of two
different systems. In the first, we study optically-induced currents in quantum
rings at zero temperature, and in the second we study a linear chain coupled at
its ends to two thermal baths with different (finite) temperatures. In
addition, we give an analytical justification for our method, based on an exact
averaging over the many-body states of the original master equations.
