In this paper, we extend the parametrization technique of Tuy et al. into a class of concave production-transportation problems with $m$ $(\ge 3)$ sources, $n$ terminals and three nonlinear variables. We develop a depth-first-search algorithm for finding a globally optimal solution of this rank three concave minimization problem and show that the algorithm is pseudo-polynonomial in the problem input length but polynomial in $m$ and $n$.