Duality for fractional transportation problem

The main purpose of this paper is to introduce a dual problem of the following fractional transportation problem min f(X) = (Σi=1^m Σj=1ⁿ cij xij)/(Σi=1^m Σj=1ⁿ dij xij) subject to (Σj=1ⁿ xij ≤ ai, i=1, ..., m; (Σi=1^m xij ≥ bj, j=1, ..., n; and xij ≥ 0, i=1, ..., m and j=1, ..., n. Weak, direct and converse duality theorems regarding the relationship between primal and dual problems are established. Finally, a numerical example is presented.