  Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type
on a compact toric symplectic manifold were investigated in \cite{Wang2} . In
this article, we consider \emph{general} toric generalized K$\ddot{a}$hler
structures of symplectic type, without requiring them to be anti-diagonal. Such
a structure is characterized by a triple $(\tau, C, F)$ where $\tau$ is a
strictly convex function defined in the interior of the moment polytope
$\Delta$ and $C, F$ are two constant anti-symmetric matrices. We prove that
underlying each such a structure is a \emph{canonical} toric K$\ddot{a}$hler
structure $I_0$ whose symplectic potential is given by this $\tau$, and when
$C=0$ the generalized complex structure $\mathbb{J}_1$ other than the
symplectic one arises from an $I_0$-holomorphic Poisson structure $\beta$ in a
\emph{novel} way not mentioned in the literature before. Conversely, given a
toric K$\ddot{a}$hler structure with symplectic potential $\tau$ and two
anti-symmetric constant matrices $C, F$, the triple $(\tau, C, F)$ then
determines a toric generalized K$\ddot{a}$hler structure of symplectic type
canonically if $F$ satisfies additionally a certain positive-definiteness
condition. In particular, if the initial toric K$\ddot{a}$hler manifold is the
standard $M_\Delta$ associated to a Delzant polytope $\Delta$, the resulting
generalized K$\ddot{a}$hler structure can be interpreted as obtained via
generalized K$\ddot{a}$hler reduction from a generalized K$\ddot{a}$hler
structure on an open subset of a complex linear space, just as in Delzant's
construction $M_\Delta$ is obtained through K$\ddot{a}$hler reduction from a
complex linear space.
