  The efficiency at maximum power (EMP) of irreversible Carnot-like heat
engines is investigated based on the weak endoreversible assumption and the
phenomenologically irreversible thermodynamics. It is found that the weak
endoreversible assumption can reduce to the conventional one for the heat
engines working at maximum power. Carnot-like heat engines are classified into
three types (linear, superlinear, and sublinear) according to different
characteristics of constitutive relations between the heat transfer rate and
the thermodynamic force. The EMPs of Carnot-like heat engines are proved to be
bounded between $\eta_C/2$ and $\eta_C/(2-\eta_C)$ for the linear type, 0 and
$\eta_C/(2-\eta_C)$ for the superlinear type, and $\eta_C/2$ and $\eta_C$ for
the sublinear type, respectively, where $\eta_C$ is the Carnot efficiency.
