  MacMahon's classical theorem on the number of boxed plane partitions has been
generalized in several directions. One way to generalize the theorem is to view
boxed plane partitions as lozenge tilings of a hexagonal region and then
generalize it by making some holes in the region and counting its tilings. In
this paper, we provide new regions whose numbers of lozenges tilings are given
by simple product formulas. The regions we consider can be obtained from
hexagons by removing structures called intrusions. In fact, we show that the
tiling generating functions of those regions under certain weights are given by
similar formulas. These give the $q$-analogue of the enumeration results.
