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

TL;DR

This paper introduces new exact and heuristic methods for efficiently scheduling jobs with release dates on identical parallel machines to minimize total weighted completion time, solving instances up to 200 jobs and 10 machines.

Contribution

It proposes a novel branch-and-price algorithm with decomposition techniques and efficient subproblem solutions for the $P|r_j| ext{sum } w_jC_j$ problem.

Findings

Exact methods solve instances with up to 200 jobs and 10 machines.

Heuristic methods achieve very low gaps on larger instances.

The approach fills a gap in recent literature on this scheduling problem.

Abstract

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.