Near-Optimal Solutions of Large-Scale Single-Machine Scheduling Problems

Single Machine Scheduling Problems with Release Dates (SMSP) concern the optimal allocation of a set of jobs on a single machine, that cannot process more than one job at a time. Each job is ready for processing at a release date and it has to be processed without interruption. The goal is to minimize the total weighted completion time of the jobs. In this paper the time-indexed formulation is considered and a new lagrangian heuristic is proposed, based on the observation that lagrangian relaxation of job constraints leads to a Weighted Stable Set problem on an Interval Graph. The problem is polinomially solvable by a dynamic programming algorithm. Computational experience is reported, showing that instances up to 400 jobs and maximum processing time 50 (around 5 millions variables) are solved in less than 40 minutes on a Personal Computer, yielding duality gaps never exceeding 3.0%. We also test a set of hard instances, built to produce bad performances, where we yield duality gaps less than 5%.