A Heuristic Based on LPT rule and a restricted B&B Method to Minimizing the Expected Makespan on a Single Machine

Considering the production and maintenance schedule simultaneously has a great attention for more than two decades. At the beginning, most of these researches treated maintenance activities as a fixed hole in the study period. Later, in the majority of them, the maintenance periods starting time were fixed but their intervals may variate. In the non-preemptive case, only few of articles that deals with the maintenance activities starting time as a decision variable. In this article, the time to starting the planed maintenance activities (PM) is a decision variable and the expected repair time is estimated according to nonhomogeneous Poisson process (NHPP). The failure function is Weibully distributed and the repair due to failure is minimal while the PM repair is perfect. Two cases were considered for the model which is aiming to minimizing the expected makespan (EMS), first, the preemptive case which was solved optimally. Secondly, a heuristic based on longest processing time rule (LPT) and a restricted branch and bound method (B&B) are provided to solve the problem for the non-preemptive case. Each of the proposed procedures uses the optimal solution properties of the preemptive case.  The optimal solution for the non-preemptive model can be reached using LPT in case of . Finally, the results showed the robustness for B&B which their lower bound obtained by the optimal solution of the preemptive case and the upper bound is the first solution obtained by LPT.