Solving the Linear Transportation Problem by Modified Vogel Method

In this chapter, we propose a modification of the Vogel Approximation Method (VAM) used to obtain near optimal solutions to linear transportation problems. This method, called Modified Vogel Method (MVM), consists of performing the row and column reduction of the cost matrix and then applying the classical Vogel method to the equivalent transportation problem with the reduced cost matrix. We prove that when no further reduction of a cost matrix is required, we do obtain an optimal solution, not an approximate one. We identify some cases when such a behavior occurs and provides rules that allow for fast new reductions and penalty calculations when needed. The method also allows us to make multiple assignments of variables. Numerical tests run on small tests show that the MVM over performs the original one in all instances while requiring comparable computing times. The tests also support the intuition that the new method provides optimal solutions almost all the time, making it a viable alternative to the classical transportation simplex.