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We consider a single-machine two-agent problem where the objective is to minimize a weighted combination of the total completion time and the total tardiness of jobs from the first agent given that no tardy jobs are allowed for the second agent. A branch-and-bound algorithm is developed to derive the optimal sequence and two simulated annealing heuristic algorithms are proposed to search for the near-optimal solutions. Computational experiments are also conducted to evaluate the proposed branch-and-bound and simulated annealing algorithms.