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The paper considers scheduling on parallel machines under the constraint thatsome pairs of jobs cannot be processed concurrently. Each job has an associatedweight, and all jobs have the same deadline. The objective is to maximise thetotal weight of on-time jobs. The problem is known to be strongly NP-hard ingeneral. A polynomial-time algorithm for scheduling unit execution time jobs ontwo machines is proposed. The performance of a broad family of approximationalgorithms for scheduling unit execution time jobs on more than two machines isanalysed. For the case of arbitrary job processing times, two integer linearprogramming formulations are proposed and compared with two formulations knownfrom the earlier literature. An iterated variable neighborhood search algorithmis also proposed and evaluated by means of computational experiments.

Maximising the total weight of on-time jobs on parallel machines subject to a conflict graph