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Makespan minimization in single-machine scheduling with step-deterioration of processing times. (English) Zbl 1095.90038
Summary: 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 $\mathrm{𝒩𝒫}$-hard in the literature. In this paper, we first show this problem to be $\mathrm{𝒩𝒫}$-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.
##### MSC:
 90B35 Scheduling theory, deterministic 90C39 Dynamic programming 90C60 Abstract computational complexity for mathematical programming problems