A highly elongated plasma is desirable in order to increase plasma pressure and energy confinement to maximize fusion power output. However, there is a limit to the maximum achievable elongation which is set by vertical instabilities driven by the n = 0 MHD mode. This limit can be increased by optimizing several parameters characterizing the plasma and the wall. The purpose of our study is to explore how and to what extent this can be done. Specifically, we extend many earlier calculations of the n = 0 mode and numerically determine scaling relations for the maximum elongation as a function of dimensionless parameters describing (1) the plasma profile ( and li), (2) the plasma shape ( and δ), (3) the wall radius (b/a) and (4) most importantly the feedback system capability parameter . These numerical calculations rely on a new formulation of n = 0 MHD theory we recently developed (Freidberg et al 2015 J. Plasma Phys. 81 515810607, Lee et al 2015 J. Plasma Phys. 81 515810608) that reduces the 2D stability problem into a 1D problem. This method includes all the physics of the ideal MHD axisymmetric instability while reducing the computation time significantly, so that many parameters can be explored during the optimization process. The scaling relations we present include the effects of the optimal triangularity and the finite aspect ratio on the maximum elongation, and can be useful for determining optimized plasma shapes in current experiments and future tokamak designs.