  We give a concrete expression of a minimal singular metric of a big line
bundle on a compact K\"ahler manifold which is the total space of a toric
bundle over a complex torus. In this class of manifolds, Nakayama constructed
examples which have line bundles admitting no Zariski-decomposition even after
any proper modifications. As an application, we discuss the Zariski-closedness
of non-nef loci and the openness conjecture of Demailly and Koll\'{a}r in this
class.
