Multi-objective ACO algorithms to minimise the makespan and the total rejection cost on BPMs with arbitrary job weights

In this paper, we investigate the batch-scheduling problem with rejection on parallel machines with non-identical job sizes and arbitrary job-rejected weights. If a job is rejected, the corresponding penalty has to be paid. Our objective is to minimise the makespan of the processed jobs and the total rejection cost of the rejected jobs. Based on the selected multi-objective optimisation approaches, two problems, P1 and P2, are considered. In P1, the two objectives are linearly combined into one single objective. In P2, the two objectives are simultaneously minimised and the Pareto non-dominated solution set is to be found. Based on the ant colony optimisation (ACO), two algorithms, called LACO and PACO, are proposed to address the two problems, respectively. Two different objective-oriented pheromone matrices and heuristic information are designed. Additionally, a local optimisation algorithm is adopted to improve the solution quality. Finally, simulated experiments are conducted, and the comparative results verify the effectiveness and efficiency of the proposed algorithms, especially on large-scale instances.