  The Lema\'itre-Toman-Bondi (LTB) models have reported to suffer from
incompatibility with cosmological observations and fine-tuning of the
observer's location. Further analysis of these issues indicates that they could
be resolved by models that are compatible with the supernova Ia data, but less
inhomogeneous than those that have been presented in the literature so far. We
study if such models exist by employing the degrees of freedom of the LTB
models in a novel manner. We discovered two scenarios which may meet the
expectations, but extensive numerical and analytical investigation showed them
inviable. We extended our studies to the $\Lambda$LTB models, which generalizes
the LTB models by including a non-zero cosmological constant $\Lambda$ in
Einsteins equations. This adds an additional degree of freedom for the earlier
scenarios and introduces a new scenario capable of meeting the expectations.
However, extensive numerical and analytical investigation reveals that
inclusion of $\Lambda$ does not enhance the viability of the models. We
identify the lack of degrees of freedom to be the reason for the unviability.
However, the method presented here can be generalized to models including more
degrees of freedom, like the Szekeres models, which have more promise to
overcome the issues in the LTB models.
