TORA Experiment. Consider the following LP (authored by E.M. Beale to demonstrate cycling):

Maximize z = 34 x1 - 20x2 + 12 x3 - 6x4 subject to 14 x1 - 8x2 - x3 + 9x4 … 0 12 x1 - 12x2 - 12 x3 + 3x4 … 0 x3 … 1 x1, x2, x3, x4 Ú 0 From TORA’s SOLVE>MODIFY menu, select Solve 1 Algebraic 1 Iterations 1 All@slack . Next, “thumb” through the successive simplex iterations using the command Next iteration (do not use All iterations , because the simplex method will then cycle indefinitely). You will notice that the starting all-slack basic feasible solution at iteration 0 will reappear identically in iteration 6. This example illustrates the occurrence of cycling in the simplex iterations and the possibility that the algorithm may never converge to the optimum solution. (It is interesting that cycling will not occur in this example if all the coefficients in this LP are converted to integer—try it!)