A note on the single-machine scheduling problem with minimum weighted completion time and maximum allowable tardiness

This paper analyzes the Smith‐heuristic for the single‐machine scheduling problem where the objective is to minimize the total weighted completion time subject to the constraint that the tradiness for any job does not exceed a prespecified maximum allowable tardiness. We identify several cases of this problem for which the Smith‐heuristic is guaranteed to lead to optimal solutions. We also provide a worst‐case analysis of the Smith‐heuristic; the analysis shows that the fractional increase in the objective function value for the Smith‐heuristic from the optimal solution is unbounded in the worst case.