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

Scheduling jobs with release dates on identical parallel machines by minimizing the total weighted completion time