In this paper we consider the nonsimultaneous multiprocessor scheduling problem, or NMSP for short. The NMSP is a makespan minimization scheduling problem which involves the non-pre-emptive assignment of independent jobs on m parallel machines with different starting times. It is well known that the longest processing time (LPT) algorithm and the modified LPT(MLPT) algorithm yield schedules with makespans bounded by 3/2 − 1/2m and 4/3 times the optimum makespan, respectively. In this paper, we show that the best known worst-case performance bound, 4/3 of the MLPT, is tight by constructing a worst-case example. Then, we employ the bin-packing heuristic algorithm called the MULTIFIT to solve the NMSP and show that the makespan of the schedule generated by the MULTIFIT algorithm is bounded by 9/7 + 2−k times the optimum makespan, where k is the selected number of the major iterations in the MULTIFIT. This worst-case bound of the MULTIFIT algorithm is, so far, the best bound for the NMSP and the tightness of the bound is still an open question.
