On the noninteger polyhedron vertices of the three-index axial transportation problem

For the three-index axial transportation polyhedron defined by the integer vector, existence of noninteger vertices was proved. In particular, the three-index $nmk$ axial transportation polyhedron having vertices with $r$ fractional components was shown to exist for and only for any number $r\in\{4,6,7,\dots,\delta (n,m,k)\}$, where $\delta(n,m,k)=\min\{n,m+k-2\}+m+k-2$, $n\geq m\geq k\geq 3$.