Improving loops and vanishing variables in long transportation problems

In this paper the following two results are presented: (1)A method which determines the optimal values of certain variables during the iterative solution process. The closer the current primal feasible solution is to the optimal solution, the greater the number of variables which may be determined. (2) For each current feasible solution (X_ij) of the given m × n transportation problem A, a feasible solution ( X ̄_ij) of an auxiliary m × m(m -1) transportation problem A ̄ is constructed. Problem A ̄ is shown to be equivalent to an m(m - 1) × m(m - 1) assignment problem with two admissible cells per column. The optimally of (X_ij) is shown to imply the optimality of ( X ̄_ij) and conversely. The best "improving loops" (including the improving loops used in MODI) of A ̄ are shown to be the best "improving loops" of A as well.