  Orbifold generalizations of the ordinary and modified melting crystal models
are introduced. They are labelled by a pair $a,b$ of positive integers, and
geometrically related to $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifolds of local
$\mathbf{CP}^1$ geometry of the $\mathcal{O}(0)\oplus\mathcal{O}(-2)$ and
$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ types. The partition functions have a
fermionic expression in terms of charged free fermions. With the aid of shift
symmetries in a fermionic realization of the quantum torus algebra, one can
convert these partition functions to tau functions of the 2D Toda hierarchy.
The powers $L^a,\bar{L}^{-b}$ of the associated Lax operators turn out to take
a special factorized form that defines a reduction of the 2D Toda hierarchy.
The reduced integrable hierarchy for the orbifold version of the ordinary
melting crystal model is the bi-graded Toda hierarchy of bi-degree $(a,b)$.
That of the orbifold version of the modified melting crystal model is the
rational reduction of bi-degree $(a,b)$. This result seems to be in accord with
recent work of Brini et al. on a mirror description of the genus-zero
Gromov-Witten theory on a $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifold of the
resolved conifold.
