  This paper examines the thermoelectric response of a dissipative quantum dot
heat engine based on the Anderson-Holstein model in two relevant operating
limits: (i) when the dot phonon modes are out of equilibrium, and (ii) when the
dot phonon modes are strongly coupled to a heat bath. In the first case, a
detailed analysis of the physics related to the interplay between the quantum
dot level quantization, the on-site Coulomb interaction and the electron-phonon
coupling on the thermoelectric performance reveals that an n-type heat engine
performs better than a p-type heat engine. In the second case, with the aid of
the dot temperature estimated by incorporating a {\it{thermometer bath}}, it is
shown that the dot temperature deviates from the bath temperature as
electron-phonon interaction becomes stronger. Consequently, it is demonstrated
that the dot temperature controls the direction of phonon heat currents,
thereby influencing the thermoelectric performance. Finally, the conditions on
the maximum efficiency with varying phonon couplings between the dot and all
the other macroscopic bodies are analyzed in order to reveal the nature of the
optimum junction.
