A solution proposal to the transportation problem with the linear fractional objective function

This paper deals with the transportation problem of minimizing the ratio of two linear functions subject to a set of linear equations and non-negativity conditions on the variables (or constraints of the classical transportation problem). In this paper, we extend the transportation problem with the linear objective function to the transportation problem with the linear fractional objective function and we propose a new algorithm in order to obtain an initial solution for this problem which is similar to Vogel approximation method in the classical transportation problem and then we construct the optimality conditions for the transportation problem with the linear fractional objective functions.