Abstract: In the field of job scheduling, multi-agent issues have been studied for many years. Most of researchers focused their attention only on two agents. However, there are more than two agents in the real-world scheduling problems. In this study, we consider a single-machine multi-agent scheduling problem with release time and maintenance activity. The objective is to minimize the first agent's total completion time given that the tardiness of jobs from the second agent does not exceed a limit and the maintenance activity from the third agent must be conducted within a specified time interval, i.e., maintenance window. A lower bound is proposed to accelerate the branch-and-bound algorithm. Computational experiments show the proposed lower bound performs well. The improvement ratio even reaches 1789%.