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 n x m x k axial transportation polyhedron having vertices with r fractional components was shown to exist for and only for any number r is an element of {4, 6, 7, ..., delta(n, m, k)}, where delta(n, m, k) = min{n, m + k - 2} + m + k - 2, n greater than or equal to m greater than or equal to k greater than or equal to 3.