A Lagrangian approach to single-machine scheduling problems with two competing agents

In this paper we develop branch and bound algorithms for a number of hard, two-agent scheduling problems, i.e., problems in which each agent has an objective function which depends on the completion times of its jobs only. We address the minimization of total weighted completion time of the first agent jobs, subject to a bound on the cost function of the second agent, which may be: (i) total weighted completion time, (ii) maximum lateness, (iii) maximum completion time. Our bounding approach is based on the fact that for all problems considered, the Lagrangian dual gives a good bound and can be solved exactly at a cheap computational cost. An extensive computational experience shows the effectiveness of this approach.