  Clustering is one of the most fundamental problem in Machine Learning.
Researchers in the field often require a lower bound on the size of the
clusters to maintain anonymity and upper bound for the ease of analysis.
Specifying an optimal cluster size is a problem often faced by scientists. In
this paper, we present a framework to obtain constant factor approximations for
some prominent clustering objectives, with lower and upper bounds on cluster
size. This enables scientists to give an approximate cluster size by specifying
the lower and the upper bounds for it. Our results preserve the lower bounds
but may violate the upper bound a little. %{GroverGD21_LBUBFL_Cocoon} to $2$.
%namely, $k$ Center (LUkC) and $k$ Median (LUkM) problem. We study the problems
when either of the bounds is uniform. We apply our framework to give the first
constant factor approximations for LUkM and its generalization, $k$-facility
location problem (LUkFL), with $\beta+1$ factor violation in upper bounds where
$\beta$ is the violation of upper bounds in solutions of upper bounded
$k$-median and $k$-facility location problems respectively. We also present a
result on LUkC with uniform upper bounds and, its generalization, lower and
(uniform) upper bounded $k$ supplier problem (LUkS). The approach also gives a
result on lower and upper bounded facility location problem (LUFL), improving
upon the upper bound violation of $5/2$ due to Gupta et al.
  We also reduce the violation in upper bounds for a special case when the gap
between the lower and upper bounds is not too small.
