Scheduling preempt-repeat version jobs on a machine subject to stochastic breakdowns to minimize quadratic earliness-tardiness penalties (i)

This article addresses the problem of scheduling n preempt-repeat version jobs with a common due date on a machine subject to stochastic breakdowns to minimize quadratic earliness-tardiness penalties. I proceed to study under the condition of the uptimes following an exponential distribution and with the method of large analysis of random variables. My main contributions are as follows. (1) An analytic expression for the second moment of the completion time about a preempt-repeat version job processed on a machine subject to stochastic breakdowns is proposed. (2) The result that the optimal solutions are V-shaped is proposed and proven.