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

TL;DR

This paper addresses scheduling on parallel machines with conflict constraints to maximize on-time weighted jobs, proposing algorithms and formulations for different job types and machine counts, including a polynomial-time solution for a specific case.

Contribution

It introduces a polynomial-time algorithm for two-machine unit-time jobs and compares new and existing ILP formulations, along with an iterated local search heuristic for general cases.

Findings

Polynomial-time algorithm for two-machine unit-time jobs.

Analysis of approximation algorithms for more than two machines.

New ILP formulations and a heuristic for arbitrary processing times.

Abstract

The paper considers scheduling on parallel machines under the constraint that some pairs of jobs cannot be processed concurrently. Each job has an associated weight, and all jobs have the same deadline. The objective is to maximise the total weight of on-time jobs. The problem is known to be strongly NP-hard in general. A polynomial-time algorithm for scheduling unit execution time jobs on two machines is proposed. The performance of a broad family of approximation algorithms for scheduling unit execution time jobs on more than two machines is analysed. For the case of arbitrary job processing times, two integer linear programming formulations are proposed and compared with two formulations known from the earlier literature. An iterated variable neighborhood search algorithm is also proposed and evaluated by means of computational experiments.