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Using effective scheduling strategies to improve turnaround time, slowdown, and utilization is an important consideration in large supercomputing environments. Since such machines have traditionally used non-preemption strategies to accommodate multiple jobs at the same time, a number of questions remain unanswered regarding the practical effect of preemptive scheduling. In this paper, we first discuss how to improve the turnaround time, slowdown, and utilization using an appropriate suspension criterion. We then develop a new multi-constraint preemptive scheduling algorithm (MCPS). Finally, we describe the experiments on a simulator with several realistic workloads. The experimental results show that MCPS is better than the conventional no-preemption strategies, and outperforms other preemptive scheduling strategies by its balanced performance.