  This paper addresses the problem of scheduling a set of jobs that are
released over the time on a set of identical parallel machines, aiming at the
minimization of the total weighted completion time. This problem, referred to
as $P|r_j|\sum w_jC_j$, is of great importance in practice, because it models a
variety of real-life applications. Despite its importance, the $P|r_j|\sum
w_jC_j$ has not received much attention in the recent literature. In this work,
we fill this gap by proposing mixed integer linear programs and a tailored
branch-and-price algorithm. Our {branch-and-price} relies on the decomposition
of an arc-flow formulation and on the use of efficient exact and heuristic
methods for solving the pricing subproblem. Computational experiments carried
out on a set of randomly generated instances prove that the proposed methods
can solve to the proven optimality instances with up to 200 jobs and 10
machines, and provide very low gaps for larger instances.
