  We study the optimal performance of a three-level quantum refrigerator using
a trade-off objective function, $\Omega$ function, which represents a
compromise between the energy benefits and the energy losses of a thermal
device. First, we optimize the performance of our refrigerator by employing a
two-parameter optimization scheme and show that the first two-terms in the
series expansion of the obtained coefficient of performance (COP) match with
those of some classical models of refrigerator. Then, in the high-temperature
limit, optimizing with respect to one parameter while constraining the other
one, we obtain the lower and upper bounds on the COP for both strong as well as
weak (intermediate) matter-field coupling conditions. In the strong
matter-field coupling regime, the obtained bounds on the COP exactly match with
the bounds already known for some models of classical refrigerators. Further
for weak matter-field coupling, we derive some new bounds on the the COP of the
refrigerator which lie beyond the range covered by bounds obtained for strong
matter-field coupling. Finally, in the parameter regime where both cooling
power and $\Omega$ function can be maximized, we compare the cooling power of
the quantum refrigerator at maximum $\Omega$ function with the maximum cooling
power.
