Makespan minimization in single-machine scheduling with step-deterioration of processing times

This paper considers a single-machine scheduling problem of minimizing the maximum completion time for a set of independent jobs. The processing time of a job is a non-linear step function of its starting time and due date. The problem is already known to be NP-hard in the literature. In this paper, we first show this problem to be NP-hard in the ordinary sense by proposing a pseudo-polynomial time dynamic programming algorithm. Then, we develop two dominance rules and a lower bound to design a branch-and-bound algorithm for deriving optimal solutions. Numerical results indicate that the proposed properties can effectively reduce the time required for exploring the solution space. Journal of the Operational Research Society (2004) 55, 247–256. doi:10.1057/palgrave.jors.2601693