  We consider the classic problem of scheduling jobs on unrelated machines so
as to minimize the weighted sum of completion times. Recently, for a small
constant $\varepsilon >0 $, Bansal et al. gave a
$(3/2-\varepsilon)$-approximation algorithm improving upon the natural barrier
of $3/2$ which follows from independent randomized rounding. In simplified
terms, their result is obtained by an enhancement of independent randomized
rounding via strong negative correlation properties.
  In this work, we take a different approach and propose to use the same
elegant rounding scheme for the weighted completion time objective as devised
by Shmoys and Tardos for optimizing a linear function subject to makespan
constraints. Our main result is a $1.21$-approximation algorithm for the
natural special case where the weight of a job is proportional to its
processing time (specifically, all jobs have the same Smith ratio), which
expresses the notion that each unit of work has the same weight. In addition,
as a direct consequence of the rounding, our algorithm also achieves a
bi-criteria $2$-approximation for the makespan objective. Our technical
contribution is a tight analysis of the expected cost of the solution compared
to the one given by the Configuration-LP relaxation - we reduce this task to
that of understanding certain worst-case instances which are simple to analyze.
