Minimizing the Total Weighted Completion Time of a Single Machine With Flexible Maintenance

This paper investigates the NP-hard problem of scheduling $n$ jobs on a single machine, where the machine must be stopped periodically for maintenance after a certain continuous working time. The objective is to minimize the total weighted completion time. Two-phase heuristic algorithms are proposed: in the first phase, initial solutions are obtained by WSPT or look-ahead (LA) algorithms; then, in the second phase, the solutions are improved through improvement procedures. Additionally, a mixed integer programming (MIP) model and a branch and bound (BAB) algorithm are developed. The results of computational experiments demonstrate that when the number of jobs is less than or equal to 20, the BAB is competitive compared with an MIP model implemented in a commercial solver because in most instances the former can obtain optimal solutions faster. The results also indicate that the LA algorithm significantly outperforms the WSPT algorithm in terms of the solution quality, and the proposed improvement procedure is evidently useful in refining the solution quality for both the WSPT and LA algorithms.