A single-machine bi-criterion scheduling problem with two agents

The multiple-agent scheduling problems have received increasing attention recently. However, most of the research focuses on studying the computational complexity of the intractable cases or examining problems with a single criterion. Often a decision maker has to decide the schedule based on multiple criteria. In this paper, we consider a single machine problem where the objective is to minimize a linear combination of the total completion time and the maximum tardiness of jobs from the first agent given that no tardy jobs are allowed for the second agent. We develop a branch-and-bound algorithm and several simulated annealing algorithms to search for the optimal solution and near-optimal solutions for the problem, respectively. Computational experiments show that the proposed branch-and-bound algorithm could solve problems of up to 24 jobs in a reasonable amount of time and the performance of the combined simulated annealing algorithm is very good with an average error percentage of less than 0.5% for all the tested cases.