Visualizing and Constructing Cycles in the Simplex Method

Most examples of cycling in the simplex method are given without explanation of how they were constructed. An exception is Beale's example built around the geometry of the dual simplex method in the plane [Beale, E. 1955. Cycling in the dual simplex method. Naval Res. Logist. Quart.2(4) 269–275]. Using this approach, we give a simple geometric explanation for a number of examples of cycling in the simplex method, including Hoffman's original example [Hoffman, A. 1953. Cycling in the Simplex Algorithm. National Bureau of Standards, Washington, D.C.]. This gives rise to a simple method for generating examples with cycles.