  In this paper we analyse the doublet-triplet splitting problem in the minimal
non-super-symmetric $SU(5)$ GUT. We take into account the full symmetry
breaking pattern with both high scale $SU(5)$ breaking and electroweak symmetry
breaking. Our analysis shows that the only phenomenologically acceptable model
has three vevs, with a strong hierarchy determined by the minimization
conditions. The amount of fine-tuning in the model is then numerically
evaluated by looking at the effect of variation of input parameters on both the
minimization conditions and the bosonic masses. Regarding the vevs as output
parameters, a large amount of fine-tuning is required in this scenario, which
is an expression of the doublet-triplet splitting problem. We show that this
problem is more general, since a model with coupled scalar sectors will in
general never realise a hierarchy in vevs. To avoid these problems we advocate
imposing the desired hierarchy in vevs as part of the theory. We argue for this
viewpoint because the $SU(5)$ breaking and electroweak symmetry breaking need
to be adjusted to each other anyway and cannot be regarded as independent
mechanisms. We suggest that not only the symmetry breaking pattern needs to be
imposed, but also the scales at which the breakings happen. We show
quantitatively that the generic theory with hierarchy imposed does not require
any fine-tuning of the free parameters which can all be natural and
perturbative as desired.
